Nonexistence of interior bubbling solutions for slightly supercritical elliptic problems

Abstract

In this paper, we consider the Neumann elliptic problem $(\mathcal{P}_{\varepsilon})$: $-\Delta u +\mu u = u^{(({n+2})/({n-2}))+\varepsilon}$, $u>0$ in Ω, ${\partial u}/{\partial \nu}=0$ on ∂Ω, where Ω is a smooth bounded domain in $\mathbb{R}^{n}$, $n\geq 4$, ε is a small positive real, and μ is a fixed positive number. We show that, in contrast with the three dimensional case, $(\mathcal{P}_{\varepsilon})$ has no solution blowing up at only interior points as ε goes to zero. The proof strategy consists in testing the equation by appropriate vector fields and then using refined asymptotic estimates in the neighborhood of bubbles, we obtain equilibrium conditions satisfied by the concentration parameters. The careful analysis of these balancing conditions allows us to obtain our results.

1 Introduction and main results

In this paper, we consider the following nonlinear elliptic equation:

$$\begin{aligned} (\mathcal{P}_{\mu,p}): \quad \textstyle\begin{cases} -\Delta u +\mu u = u^{p},& u>0 \text{ in } \Omega, \\ \frac{\partial u}{\partial \nu}=0, & \text{on } \partial \Omega, \end{cases}\displaystyle \end{aligned}$$

where $1 < p < \infty$, Ω is a smooth bounded domain in $\mathbb{R}^{n}$, $n\geq 3$, and μ is a positive number.

The interest in Problem $(\mathcal{P}_{\mu,p})$ grew up from the fact that it models several phenomena in applied sciences. For example it can be seen as a steady-state problem for parabolic problems in chemotaxis, e.g., Keller–Segel model [13], or for the shadow system of the Gierer–Meinhardt system in biological pattern formation [10, 16].

Many works have been devoted to problem $(\mathcal{P}_{\mu,p})$. It is well known that the situation depends on both the parameter μ and the exponent p. When μ is small and p is subcritical, i.e., $1 < p < ({n+2})/({n-2})$, the only solution is the constant one [13]. For large μ and p subcritical, it is known that solutions exist and concentrate at one or several points located in the interior of the domain, on the boundary, or some of them on the boundary and others in the interior (see the review in [17]). In the critical case, i.e., $p = ({n+2})/({n-2})$, when μ is small, $n=3$ and Ω is convex, the only solution is the constant one [31, 32]. However, for $n\in \{4,5,6\}$ and μ small, nonconstant solutions exist (see [4] when Ω is a ball and [25, 30] for general domains). When μ is large, nonconstant solutions also exist (see [1, 26]) and, as in the subcritical case, solutions blow up at one or several boundary points as μ goes to infinity (see [2, 3, 11, 14, 15, 18, 21, 27–29]). The question of the existence of interior blow-up points is still open. However, in contrast with the subcritical case, we know that at least one point must lie on the boundary [22]. In the supercritical case, very little is known. When Ω is a ball, the uniqueness of the radial solution is proved for small μ [12]. For a general smooth bounded domain and a slightly supercritical p, i.e., $p = (({n+2})/({n-2})) +\varepsilon$, where $\varepsilon >0$ $\varepsilon \to 0$, a single boundary bubble solution exists for fixed $\mu >0$ and $n\geq 4$ [9, 24]. Furthermore, a single interior bubble solution has been constructed in [23] for $n=3$. Notice that the slightly supercritical pure Neumann problem, that is, $\mu =0$ and ε is a small positive real, has been studied recently in [19], and the authors proved the existence and multiplicity of bubbling solutions in a ball. In this paper, we focus on a new phenomenon, which is the nonexistence of interior bubbling solutions for slightly supercritical case when $n\geq 4$. Thus, in what remains of this paper, we consider the slightly supercritical problem

$$\begin{aligned} (\mathcal{P}_{\varepsilon}):\quad \textstyle\begin{cases} -\Delta u +\mu u = u^{p+\varepsilon}, & u>0 \text{ in } \Omega, \\ \frac{\partial u}{\partial \nu}=0, &\text{on } \partial \Omega, \end{cases}\displaystyle \end{aligned}$$

where ε is a small positive real, Ω is a smooth bounded domain in $\mathbb{R}^{n}$, $n\geq 4$, μ is a positive fixed number, and $p+1=2n/(n-2)$ is the critical Sobolev exponent for the embedding $H^{1}(\Omega ) \to L^{q}(\Omega )$.

Before we state our main result, we need to introduce some notation. Let us define the following family of functions called bubbles:

$$\begin{aligned} \delta _{a,\lambda } (x) = c_{0} \frac{\lambda ^{(n-2)/2}}{(1+\lambda ^{2} \vert x-a \vert ^{2})^{(n-2)/2}},\quad \lambda >0, a, x \in \mathbb{R}^{n}, c_{0}=\bigl(n(n-2)\bigr)^{(n-2)/4}, \end{aligned}$$

(1)

which are the only solutions to the problem [7]

$$\begin{aligned} -\Delta u= u^{(n+2)/(n-2)},\quad u > 0 \text{ in } \mathbb{R}^{n}. \end{aligned}$$

We first exclude, in contrast with the three dimensional case [23], the existence of solutions which blow up at a single point lying in the interior of the domain as ε goes to 0. Notice that if $(u_{\varepsilon })$ is a sequence of nonconstant solutions to $(\mathcal{P}_{\varepsilon})$, then there are several and equivalent ways to define blow-up points of $(u_{\varepsilon })$. For example, $a\in \bar{\Omega }$ will be said to be a blow-up point of $(u_{\varepsilon })$ if

$$\begin{aligned} \liminf_{r\to 0}\limsup_{\varepsilon \to 0} \int _{B(a,r)\cap \Omega } \vert \nabla u_{\varepsilon } \vert ^{2} \biggl(\text{or} \int _{B(a,r)\cap \Omega } | u_{\varepsilon }^{\frac{2n}{n-2}} \biggr) >0. \end{aligned}$$

Our first result is the following.

Theorem 1.1

Let $n\geq 4$ and let μ be a fixed positive number. Then $(\mathcal{P}_{\varepsilon})$ has no solution $u_{\varepsilon }$ that blows up, as $\varepsilon \to 0$, at a single interior point in the sense that

$$\begin{aligned} \Vert u_{\varepsilon }- \delta _{a_{\varepsilon },\lambda _{\varepsilon } } \Vert _{H^{1}(\Omega )} \to 0, \end{aligned}$$

with $a_{\varepsilon } \to \overline{a} \in \Omega$ and $\lambda _{\varepsilon } \to \infty$ as $\varepsilon \to 0$.

To study the question of interior blow-up points without any assumption on the number of these points, we need to get some information about such possible solutions. This is the goal of the following result.

Theorem 1.2

Let $n\geq 4$ and μ be a fixed positive number. Let $(u_{\varepsilon})$ be a sequence of solutions of $(\mathcal{P}_{\varepsilon})$ such that

$$\begin{aligned} u_{\varepsilon }= \sum_{i=1}^{N} \delta _{a_{i,\varepsilon }, \lambda _{i,\varepsilon } } + v_{\varepsilon } \end{aligned}$$

with $N\geq 2$, $\| v_{\varepsilon }\|_{H^{1}(\Omega )} \to 0$, $\lambda _{i,\varepsilon } \to \infty$ and $a_{i,\varepsilon }\to \overline{a}_{i}\in \Omega$ as $\varepsilon \to 0$ for all $i \in \{1,\ldots, N\}$.

Then the following facts hold:

• (i) For each $j\in \{1,\ldots, N\}$ satisfying $\frac{\lambda _{j, \varepsilon }}{\lambda _{\mathrm{min}, \varepsilon }} \not \to \infty$ as $\varepsilon \to 0$, there exists $k\ne j$ such that $|a_{j,\varepsilon }- a_{k,\varepsilon }|\to 0$ as $\varepsilon \to 0$.

• (ii) In addition, if $n \geq 5$, then $\frac{\lambda _{\mathrm{max}, \varepsilon }}{\lambda _{\mathrm{min}, \varepsilon }} \to \infty$ as $\varepsilon \to 0$.

Theorem 1.2 allows us to generalize Theorem 1.1. More precisely, our next result shows the nonexistence of solutions with two or three interior blow-up points.

Theorem 1.3

Let $n\geq 5$ and $N =2$ or $n\geq 6$ and $N=3$. Let μ be a fixed positive number. Then $(\mathcal{P}_{\varepsilon})$ has no solution $u_{\varepsilon }$ that blows up, as $\varepsilon \to 0$, at N interior points $a_{1,\varepsilon }$,…, $a_{N,\varepsilon }$ in the sense that

$$\begin{aligned} \Biggl\Vert u_{\varepsilon }- \sum_{i=1}^{N} \delta _{a_{i,\varepsilon }, \lambda _{i,\varepsilon } } \Biggr\Vert _{H^{1}(\Omega )} \to 0 \end{aligned}$$

with, for each $i\in \{1,\ldots, N\}$, $a_{i,\varepsilon }\to \overline{a}_{i}\in \Omega$ and $\lambda _{i,\varepsilon } \to \infty$ as $\varepsilon \to 0$.

In the case of N interior blow-up points, with $N\geq 4$, the situation becomes more delicate. However, we note that Theorem 1.2 easily gives the following partial nonexistence result.

Corollary 1.4

Let $n\geq 4$, $N\geq 4$, and μ be a fixed positive number. Then $(\mathcal{P}_{\varepsilon})$ has no solution $u_{\varepsilon }$ such that

$$\begin{aligned} u_{\varepsilon }= \sum_{i=1}^{N} \delta _{a_{i,\varepsilon }, \lambda _{i,\varepsilon } } + v_{\varepsilon } \end{aligned}$$

with $\| v_{\varepsilon }\|_{H^{1}(\Omega )} \to 0$, $\lambda _{i,\varepsilon } \to \infty$, $a_{i,\varepsilon }\to \overline{a}_{i}\in \Omega$ as $\varepsilon \to 0$ for all $i\in \{1,\ldots, N\}$, and one of the following two conditions holds:

• (i) $n \geq 5$ and $\frac{\lambda _{\mathrm{max}, \varepsilon }}{\lambda _{\mathrm{min}, \varepsilon }} \not \to \infty$ as $\varepsilon \to 0$,

• (ii) $n\geq 4$ and there exists $j\in \{1,\ldots, N\}$ satisfying $\frac{\lambda _{j, \varepsilon }}{\lambda _{\mathrm{min}, \varepsilon }} \not \to \infty$ as $\varepsilon \to 0$ and $|a_{j,\varepsilon }- a_{k,\varepsilon }|\geq C>0$ as $\varepsilon \to 0$ for all $k\in \{1,\ldots, N\}$ with $k\ne j$.

To prove our results, we test the equation by appropriate vector fields and then, using refined asymptotic estimates in the neighborhood of bubbles, we obtain equilibrium conditions satisfied by the concentration parameters. The careful analysis of these balancing conditions allows us to obtain our results.

The remainder of the paper is organized as follows: in Sect. 2 we give some basic tools that we use in our proofs. In Sect. 3 we provide an accurate estimate of the gradient terms in the neighborhood of bubbles. Section 4 is devoted to the proof of our results. In Sect. 5, we discuss some future perspectives. Finally, we collect in Sect. 5 some useful estimates needed in this paper

2 Some basic tools

For v, $w \in H^{1}(\Omega )$, we set

$$\begin{aligned} \langle v, w \rangle = \int _{\Omega }\nabla v \cdot \nabla w + \mu \int _{\Omega }v w, \quad \Vert v \Vert ^{2} = \langle v, v \rangle. \end{aligned}$$

(2)

Throughout the sequel we assume that $n\geq 3$, $u_{\varepsilon }$ is a sequence of solutions of $(\mathcal{P}_{\varepsilon})$ written in the form

$$\begin{aligned} u_{\varepsilon }= \sum_{i=1}^{N} \delta _{a_{i,\varepsilon }, \lambda _{i,\varepsilon } } + v_{\varepsilon } \end{aligned}$$

with $N\geq 1$, $\| v_{\varepsilon }\|_{H^{1}(\Omega )} \to 0$, $\lambda _{i,\varepsilon } \to \infty$ and $a_{i,\varepsilon }\to \overline{a}_{i}\in \Omega$ as $\varepsilon \to 0$ for all i.

To simplify the notation, throughout the sequel we set $\delta _{i}=\delta _{a_{i,\varepsilon },\lambda _{i,\varepsilon }}$, $a_{i}=a_{i,\varepsilon }$, and $\lambda _{i}=\lambda _{i,\varepsilon }$.

We know that there is a unique way to choose $a_{i,\varepsilon }$, $\lambda _{i,\varepsilon }$, and $v_{\varepsilon }$ such that

$$\begin{aligned} u_{\varepsilon }= \sum_{i=1}^{N} \alpha _{i,\varepsilon }\delta _{a_{i, \varepsilon },\lambda _{i,\varepsilon } } + v_{\varepsilon } \end{aligned}$$

(3)

with

$$\begin{aligned} \textstyle\begin{cases} \alpha _{i,\varepsilon }\to 1, \qquad a_{i,\varepsilon } \in \Omega,\qquad a_{i, \varepsilon }\to \overline{a}_{i}\in \Omega, \qquad \lambda _{i, \varepsilon }\to \infty, \\ \varepsilon _{ij}:= (\frac{\lambda _{i}}{\lambda _{j}} + \frac{\lambda _{j}}{\lambda _{i}} +\lambda _{i}\lambda _{j} \vert a_{i}-a_{j} \vert ^{2} )^{(2-n)/2}\to 0, \\ v_{\varepsilon }\to 0 \quad \text{in } H^{1}(\Omega ), v_{ \varepsilon }\in E_{a,\lambda}, \end{cases}\displaystyle \end{aligned}$$

(4)

where, for any $(a,\lambda )\in \Omega ^{N}\times (0,\infty )^{N}$, $E_{a,\lambda}$ denotes

$$\begin{aligned} E_{a,\lambda}= {}&\biggl\{ v \in H^{1}(\Omega ): \int _{\Omega} \nabla v \cdot \nabla \delta _{i} = \int _{\Omega} \nabla v \cdot \nabla \frac{\partial \delta _{i}}{\partial \lambda _{i}} = \int _{ \Omega} \nabla v \cdot \nabla \frac{\partial \delta _{i}}{\partial a_{i}^{j}} = 0, \\ & \forall 1\leq i \leq N, \forall 1 \leq j \leq n \biggr\} . \end{aligned}$$

For the proof of this fact, see [20]. In what follows, we always assume that $u_{\varepsilon }$ is written as in (3) and (4). We start by proving the following crucial lemma.

Lemma 2.1

Let $n \geq 3$. For all $j\in \{1,\ldots, N\}$, it holds

$$\begin{aligned} \varepsilon \ln \lambda _{j} \to 0 \quad\textit{as } \varepsilon \to 0. \end{aligned}$$

Proof

Multiplying $(\mathcal{P}_{\varepsilon})$ by $\delta _{i}$ and integrating on Ω, we obtain

$$\begin{aligned} &{ -}\sum_{j=1}^{N} \alpha _{j} \int _{\Omega }\Delta \delta _{j} \delta _{i} - \int _{\Omega }\Delta v_{\varepsilon }\delta _{i} +\mu \sum_{j=1}^{N} \alpha _{j} \int _{\Omega }\delta _{j}\delta _{i} + \mu \int _{\Omega }v_{\varepsilon }\delta _{i} \\ &\quad = \int _{\Omega } \Biggl( \sum_{j=1}^{N} \alpha _{j}\delta _{j}+v_{\varepsilon } \Biggr)^{p+ \varepsilon }\delta _{i}. \end{aligned}$$

(5)

Using Lemma 6.6, we have

$$\begin{aligned} &- \int _{\Omega }\Delta \delta _{j} \delta _{i}= \int _{\Omega } \delta _{j}^{p}\delta _{i}=O(\varepsilon _{ij})=o(1)\quad \forall j \neq i, \\ &- \int _{\Omega }\Delta \delta _{i} \delta _{i}= \int _{\Omega } \delta _{i}^{p+1}= \int _{\mathbb{R}^{n}} \delta _{i}^{p+1}- \int _{ \mathbb{R}^{n}\setminus \Omega } \delta _{i}^{p+1}= S_{n}+O \biggl( \frac{1}{\lambda _{i}^{n}} \biggr), \end{aligned}$$

where

$$\begin{aligned} S_{n}=c_{0}^{p+1} \int _{\mathbb{R}^{n}}\frac{dx}{(1+ \vert x \vert ^{2})^{n}}. \end{aligned}$$

(6)

Now, since $\partial u_{\varepsilon }/\partial \nu =0$, we observe that

$$\begin{aligned} - \int _{\Omega }\Delta v_{\varepsilon }\delta _{i} = \int _{\Omega } \nabla v_{\varepsilon }\nabla \delta _{i}- \int _{\partial \Omega } \frac{\partial v_{\varepsilon }}{\partial \nu}\delta _{i} = \sum _{j=1}^{N} \alpha _{j} \int _{\partial \Omega } \frac{\partial \delta _{j}}{\partial \nu}\delta _{i} = O \Biggl(\sum_{k=1}^{N} \frac{1}{\lambda _{k}^{n-2}} \Biggr) = o(1), \end{aligned}$$

and using Lemma 6.6, we get

$$\begin{aligned} \int _{\Omega }\delta _{j}\delta _{i} + \int _{\Omega }\delta _{i}^{2} + \int _{\Omega } \vert v_{\varepsilon } \vert \delta _{i} = O \biggl( \varepsilon _{ij} + \frac{ \ln ^{\sigma _{n}} \lambda _{i} }{\lambda _{i}^{\min (n-2,2)} } + \Vert v_{\varepsilon } \Vert \biggl( \int _{\Omega }\delta _{i}^{2n/(n+2)} \biggr)^{(n+2)/(2n)} \biggr)=o(1), \end{aligned}$$

where $\sigma _{4} = 1$ and $\sigma _{n} = 0$ if $n \neq 4$.

It remains to estimate the right-hand side of (5). Using Lemma 6.1, we get

$$\begin{aligned} &\int _{\Omega } \Biggl(\sum_{j = 1}^{N} \alpha _{j}\delta _{j} + v_{ \varepsilon } \Biggr)^{p+\varepsilon }\delta _{i} \\ &\quad= \int _{\Omega }( \alpha _{i}\delta _{i})^{p+\varepsilon } \delta _{i} + O \biggl(\sum_{j \neq i} \int \bigl(\delta _{j}^{p+\varepsilon }\delta _{i} + \delta _{i}^{p+ \varepsilon }\delta _{j} \bigr) + \int \bigl( \vert v_{\varepsilon } \vert ^{p+ \varepsilon }\delta _{i} + \delta _{i} ^{p+\varepsilon } \vert v_{ \varepsilon } \vert \bigr) \biggr). \end{aligned}$$

(7)

Concerning the first integral on the right-hand side of (7), it holds

$$\begin{aligned} \int _{\Omega }\delta _{i}^{p+\varepsilon }\delta _{i}&=c_{0}^{p+1+ \varepsilon } \int _{\Omega } \frac{\lambda _{i}^{n+\varepsilon \frac{n-2}{2}}\,dx}{(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2})^{n+\varepsilon \frac{n-2}{2}}} \\ & = c_{0}^{p+1+\varepsilon } \int _{\mathbb{R}^{n}} \frac{\lambda _{i}^{n+\varepsilon \frac{n-2}{2}}dx}{(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2})^{n+\varepsilon \frac{n-2}{2}}}-c_{0}^{p+1+ \varepsilon } \int _{\mathbb{R}^{n}\setminus \Omega } \frac{\lambda _{i}^{n+\varepsilon \frac{n-2}{2}}\,dx}{(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2})^{n+\varepsilon \frac{n-2}{2}}}. \end{aligned}$$

But we have

$$\begin{aligned} &\int _{\mathbb{R}^{n}} \frac{ c_{0}^{p+1+\varepsilon } \lambda _{i}^{n+\varepsilon \frac{n-2}{2}} }{(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2})^{n+\varepsilon \frac{n-2}{2}}} \,dx = \lambda _{i}^{\varepsilon \frac{n-2}{2}}c_{0}^{\varepsilon } \int _{\mathbb{R}^{n}} \frac{c_{0}^{p+1}\,dx}{(1+ \vert x \vert ^{2})^{n+\varepsilon \frac{n-2}{2}}} = \lambda _{i}^{\varepsilon \frac{n-2}{2}} \bigl(S_{n}+O(\varepsilon ) \bigr), \\ &\int _{\mathbb{R}^{n}\setminus \Omega } \frac{\lambda _{i}^{n+\varepsilon \frac{n-2}{2}}\,dx}{(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2})^{n+\varepsilon \frac{n-2}{2}}} \leq \lambda _{i}^{\varepsilon \frac{n-2}{2}}c \int _{\mathbb{R}^{n} \setminus \Omega } \frac{\lambda _{i}^{n} \,dx}{(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2})^{n}} \leq c \frac{\lambda _{i}^{\varepsilon \frac{n-2}{2}}}{\lambda _{i}^{n}}. \end{aligned}$$

For the second integral on the right-hand side of (7), using the fact that $\delta _{i}^{\varepsilon }\leq c\lambda _{i}^{\varepsilon (n-2)/2}$, we get for $j\neq i$

$$\begin{aligned} & \int _{\Omega } \bigl( \delta _{j}^{p+\varepsilon }\delta _{i} + \delta _{j} \delta _{i} ^{p+\varepsilon } \bigr) \leq c \lambda _{i}^{ \varepsilon \frac{n-2}{2}} \int _{\Omega } \bigl( \delta _{j}^{p+ \varepsilon }\delta _{i}^{1-\varepsilon } + \delta _{j} \delta _{i} ^{p} \bigr)\,dx \leq c \lambda _{i}^{\varepsilon \frac{n-2}{2}} \varepsilon _{ij}^{1- \varepsilon }=o \bigl( \lambda _{i}^{\varepsilon \frac{n-2}{2}} \bigr), \\ & \int _{\Omega } \bigl( \vert v_{\varepsilon } \vert ^{p+\varepsilon }\delta _{i} + \vert v_{\varepsilon } \vert \delta _{i} ^{p+\varepsilon } \bigr) \leq c \lambda _{i}^{\varepsilon \frac{n-2}{2}} \int _{\Omega } \bigl( \vert v_{ \varepsilon } \vert ^{p+\varepsilon }\delta _{i}^{1-\varepsilon } + \vert v_{ \varepsilon } \vert \delta _{i}^{p} \bigr) \leq c \Vert v_{\varepsilon } \Vert \lambda _{i}^{\varepsilon \frac{n-2}{2}}=o \bigl( \lambda _{i}^{ \varepsilon \frac{n-2}{2}} \bigr). \end{aligned}$$

Combining the above estimates, we obtain

$$\begin{aligned} S_{n}+o(1) = S_{n}\lambda _{i}^{\varepsilon \frac{n-2}{2}} +o \bigl( \lambda _{i}^{\varepsilon \frac{n-2}{2}} \bigr) = \bigl( S_{n}+o(1) \bigr) \lambda _{i}^{\varepsilon \frac{n-2}{2}}. \end{aligned}$$

Hence $\lambda _{i}^{\varepsilon (n-2)/2} = 1 + o(1)$, which completes the proof of the lemma. □

Notice that since $|u_{\varepsilon }|_{\infty}$ is of the same order as $\lambda _{\mathrm{max}}^{(n-2)/2}$, Lemma 2.1 implies the following important remark.

Remark 2.2

There is $\varepsilon _{0}>0$ such that, for $\varepsilon \in (0,\varepsilon _{0})$, we have

$$\begin{aligned} \vert u_{\varepsilon } \vert _{\infty}^{\varepsilon }\leq C\quad \text{and}\quad \vert v_{\varepsilon } \vert _{\infty}^{\varepsilon }\leq C, \end{aligned}$$

where C is a positive constant independent of ε.

Now, we are going to estimate the $v_{\varepsilon }$-part in (3). To this aim, we need to prove the coercivity of the following quadratic form:

$$\begin{aligned} Q(v)= \int _{\Omega } \vert \nabla v \vert ^{2} +\mu \int _{\Omega }v^{2} -p\sum_{i=1}^{N} \int _{\Omega }\delta _{a_{i},\lambda _{i}}^{p-1}v^{2},\quad v \in E_{a,\lambda }. \end{aligned}$$

(8)

In the case of Dirichlet boundary conditions, this kind of coercivity is proved by Bahri [5]. Such a result was adapted in [20] to our case when the concentration points do not approach each other. What we need here is a result which holds even if the points are close to each other. More precisely, we will give some general formulae for future use. To this aim, let

$$\begin{aligned} \mathcal{V}(N,\varepsilon,\eta ):={}& \bigl\{ (\alpha,a,\lambda ) \in (0, \infty )^{N}\times \Omega ^{N} \times (0,\infty )^{N}: \vert \alpha _{i} - 1 \vert < \eta; \varepsilon \ln \lambda _{i} < \eta; \\ & \lambda _{i} d(a_{i},\partial \Omega ) > \eta ^{-1} \forall i; \varepsilon _{ij} < \eta \ \forall i \neq j \bigr\} , \end{aligned}$$

(9)

where $N \in \mathbb{N}$ and $\eta > 0$ is a small parameter.

Proposition 2.3

Let $n \geq 3$ and $(\alpha,a,\lambda ) \in \mathcal{V}(N,\varepsilon,\eta )$. Then there exists $\rho _{0} > 0$ such that

$$\begin{aligned} Q(v) \geq \rho _{0} \Vert v \Vert ^{2}\quad \forall v\in E_{a,\lambda }. \end{aligned}$$

Proof

Let $v \in E_{a,\lambda }$ and $v_{1}$ be its projection onto $H^{1}_{0}(\Omega )$ defined by

$$\begin{aligned} \Delta v_{1}=\Delta v \quad\text{in } \Omega, \qquad v_{1}=0 \quad\text{on } \partial \Omega \end{aligned}$$

and define $v_{2}=v-v_{1}$. It is easy to obtain

$$\begin{aligned} & \int _{\Omega }\nabla v_{1}\nabla v_{2} = - \int _{\Omega }v_{1} \Delta v_{2} + \int _{\partial \Omega }v_{1} \frac{\partial v_{2}}{\partial \nu}=0, \end{aligned}$$

(10)

$$\begin{aligned} & \int _{\Omega } \vert \nabla v \vert ^{2}= \int _{\Omega } \vert \nabla v_{1} \vert ^{2} + \int _{\Omega } \vert \nabla v_{2} \vert ^{2}. \end{aligned}$$

(11)

For $y\in \Omega$, we denote by $d_{y}:= d(y,\partial \Omega )$. Since $v_{2}$ is a harmonic function in Ω, we see that

$$\begin{aligned} \bigl\vert v_{2}(y) \bigr\vert = \biggl\vert \int _{\partial \Omega } \frac{\partial G_{0}}{\partial \nu}(x,y) v_{2}(x) \,dx \biggr\vert \leq c \int _{\partial \Omega } \frac{ \vert v_{2}(x) \vert }{ \vert x - y \vert ^{n-1} } \,dx \leq c { \Vert v_{2} \Vert _{H^{1}}} / { d_{y} ^{ (n-2)/2 } }, \end{aligned}$$

(12)

where $G_{0}$ is the Green’s function of the Laplace operator with Dirichlet boundary conditions and where we have used, in the last inequality, Holder’s inequality and Lemma 6.7. In the same way, we have

$$\begin{aligned} \bigl\vert \nabla v_{2}(y) \bigr\vert \leq c \Vert v_{2} \Vert _{H^{1}} / { d_{y} ^{ n/2 } } \quad\forall y \in \Omega. \end{aligned}$$

(13)

Next, for $1\leq i\leq N$, taking

$$\begin{aligned} \psi _{i} \in \biggl\{ \delta _{i}, \lambda _{i} \frac{\partial \delta _{i}}{\partial \lambda _{i}}, \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial (a_{i})_{j}}, 1\leq j \leq n \biggr\} \end{aligned}$$

(14)

and $B_{i}:= B(a_{i}, d_{i} /2)$, we observe that, for each $y \in B_{i}$, it holds that $d_{i} / 2 \leq d_{y} \leq 2 d_{i}$. Therefore, we get

$$\begin{aligned} \int _{\Omega }\nabla v_{1}\nabla \psi _{i} &= \int _{\Omega }\nabla v \nabla \psi _{i} - \int _{\Omega }\nabla v_{2}\nabla \psi _{i} \\ & = - \int _{ B_{i} } \nabla v_{2}\nabla \psi _{i} + O \biggl( \frac{1}{\lambda _{i}^{(n-2)/2}} \int _{\Omega \setminus B_{i} } \vert \nabla v_{2} \vert \frac{1}{ \vert y - a_{i} \vert ^{n-1} } \,dy \biggr) \\ & = O \biggl( \frac{ \Vert v_{2} \Vert _{H^{1}} }{ d_{i} ^{n/2} } \int _{ B_{i} } \vert \nabla \psi _{i} \vert + \frac{ \Vert v_{2} \Vert _{H^{1}}}{( \lambda _{i} d_{i} )^{(n-2)/2}} \biggr) = O \biggl( \frac{ \Vert v_{2} \Vert _{H^{1}}}{ ( \lambda _{i} d_{i} ) ^{(n-2)/2}} \biggr), \end{aligned}$$

(15)

where we have used (13), the fact that $\int _{ B_{i} }|\nabla \psi _{i}|= O ({ d_{i} } / {\lambda _{i}^{(n-2)/2}} )$ (for the first integral), and Holder’s inequality (for the second integral). Now, we write

$$\begin{aligned} Q(v) = Q_{0}(v_{1})+ \int _{\Omega } \vert \nabla v_{2} \vert ^{2} +\mu \int _{ \Omega }v^{2} -p\sum_{i=1}^{N} \int _{\Omega }\delta _{i}^{p-1}(2v_{1} +v_{2})v_{2}, \end{aligned}$$

(16)

where

$$\begin{aligned} Q_{0}(v_{1})= \int _{\Omega } \vert \nabla v_{1} \vert ^{2} -p\sum_{i=1}^{N} \int _{ \Omega }\delta _{i}^{p-1}v_{1}^{2}. \end{aligned}$$

However, using (12), we obtain

$$\begin{aligned} &\int _{\Omega }\delta _{i}^{p-1}\bigl(2 \vert v_{1} \vert + \vert v_{2} \vert \bigr) \vert v_{2} \vert \\ &\quad = \int _{ B_{i} } \delta _{i}^{p-1}\bigl(2 \vert v_{1} \vert + \vert v_{2} \vert \bigr) \vert v_{2} \vert + \int _{\Omega \setminus B_{i} } \delta _{i}^{p-1}\bigl(2 \vert v_{1} \vert + \vert v_{2} \vert \bigr) \vert v_{2} \vert \\ &\quad\leq c { \Vert v_{2} \Vert } \bigl( \Vert v_{1} \Vert _{H^{1}}+ \Vert v_{2} \Vert _{H^{1}}\bigr) \biggl[ \frac{ 1 }{ d_{i} ^{ (n-2)/2 }} \biggl( \int _{ B_{i} } \delta _{i}^{ \frac{8n}{n^{2}-4}} \biggr)^{\frac{n+2}{2n}}+ \biggl( \int _{\Omega \setminus B_{i} } \delta _{i}^{p+1} \biggr)^{\frac{2}{n}} \biggr] \\ &\quad=o\bigl( \Vert v_{1} \Vert _{H^{1}}^{2}+ \Vert v_{2} \Vert _{H^{1}}^{2}\bigr). \end{aligned}$$

(17)

To proceed further, let

$$\begin{aligned} \tilde{v}_{1}=v_{1} \quad\text{in } \Omega,\qquad \tilde{v}_{1}=0 \quad\text{in } \mathbb{R}^{n}\setminus \Omega. \end{aligned}$$

Clearly, $\tilde{v}_{1}\in \mathcal{D}^{1,2}(\mathbb{R}^{n})$. Now, we write

$$\begin{aligned} & \tilde{v}_{1}=\sum_{i=1}^{N(n+2)} \gamma _{i}\psi _{i} + \overline{v}_{1} \quad\text{where } \\ & \psi _{i} \in \biggl\{ \delta _{k}, \lambda _{k} \frac{\partial \delta _{k}}{\partial \lambda _{k}}, \frac{1}{\lambda _{k}} \frac{\partial \delta _{k}}{\partial (a_{k})_{j}}, 1\leq k \leq N ; 1\leq j \leq n \biggr\} \quad \text{and}\quad \int _{ \mathbb{R}^{n}} \nabla \overline{v}_{1} \nabla \psi _{i} =0 \quad\forall i. \end{aligned}$$

Using (15), we obtain

$$\begin{aligned} \int _{\mathbb{R}^{n}} \nabla \tilde{v}_{1} \nabla \psi _{i} = c \gamma _{i} + o\Bigl( \sum \gamma _{j} \Bigr) = \int _{\Omega }\nabla v_{1} \nabla \psi _{i} = o\bigl( \Vert v_{2} \Vert _{H^{1}}\bigr) \quad\forall i. \end{aligned}$$

(18)

This implies that

$$\begin{aligned} \int _{\mathbb{R}^{n}} \vert \nabla \overline{v}_{1} \vert ^{2}= \int _{ \mathbb{R}^{n}} \vert \nabla \tilde{v}_{1} \vert ^{2} + o\bigl( \Vert v_{2} \Vert _{H^{1}}^{2} \bigr). \end{aligned}$$

(19)

Using (19), we get

$$\begin{aligned} Q_{0}(v_{1})&= \int _{\mathbb{R}^{n}} \vert \nabla \tilde{v}_{1} \vert ^{2} - p \sum \int _{\mathbb{R}^{n}} \delta _{i}^{p-1} \tilde{v}_{1}^{2} \\ &= \int _{\mathbb{R}^{n}} \vert \nabla \overline{v}_{1} \vert ^{2} - p\sum \int _{ \mathbb{R}^{n}} \delta _{i}^{p-1} \overline{v}_{1}^{2} +o\bigl( \Vert \nabla \overline{v}_{1} \Vert _{L^{2}}^{2} + \Vert v_{2} \Vert _{H^{1}}^{2}\bigr). \end{aligned}$$

(20)

Combining (18) and Proposition 3.1 of [5], we obtain

$$\begin{aligned} Q_{0}(v_{1}) &\geq \rho \int _{\mathbb{R}^{n}} \vert \nabla \overline{v}_{1} \vert ^{2} +o\bigl( \Vert \nabla \overline{v}_{1} \Vert _{L^{2}}^{2} + \Vert v_{2} \Vert _{H^{1}}^{2}\bigr) \\ &\geq \frac{\rho}{2} \int _{\mathbb{R}^{n}} \vert \nabla \tilde{v}_{1} \vert ^{2} +o\bigl( \Vert v_{2} \Vert _{H^{1}}^{2} \bigr) = \frac{\rho}{2} \int _{\Omega } \vert \nabla v_{1} \vert ^{2} +o\bigl( \Vert v_{2} \Vert _{H^{1}}^{2} \bigr). \end{aligned}$$

(21)

Therefore (21) and (16) imply that

$$\begin{aligned} Q (v) & \geq \frac{\rho}{2} \int _{\Omega } \vert \nabla v_{1} \vert ^{2} + \int _{ \Omega } \vert \nabla v_{2} \vert ^{2}+ \mu \int _{\Omega }v^{2}+o\bigl( \Vert v_{1} \Vert ^{2} + \Vert v_{2} \Vert _{H^{1}}^{2} \bigr) \geq C \Vert v \Vert ^{2}, \end{aligned}$$

which completes the proof of Proposition 2.3. □

Next, we are going to estimate the norm of $v_{\varepsilon }$-part of $u_{\varepsilon }$ when ε goes to zero.

Proposition 2.4

Let $n \geq 3$ and let $v_{\varepsilon }$ be the remainder term defined in (3). Then there is $\varepsilon _{0}>0$ such that, for $\varepsilon \in (0,\varepsilon _{0})$, the following fact holds:

$$\begin{aligned} & \Vert v_{\varepsilon } \Vert \leq C R (\varepsilon,a,\lambda )\quad \textit{with} \\ & R (\varepsilon,a,\lambda ):= \varepsilon + \sum_{i=1}^{N} \frac{1}{ ( \lambda _{i} d_{i} )^{\frac{n-2}{2}}} + \sum_{1\leq i,j \leq N, j \neq i } \bigl(\varepsilon _{ij} ^{\frac{n+2}{2(n-2)}} \bigl(\ln \varepsilon _{ij}^{-1} \bigr)^{\frac{n+2}{2n}}+ \varepsilon _{ij} \bigr) \\ &\phantom{R (\varepsilon,a,\lambda ):=} + \sum_{ i = 1 }^{N} \textstyle\begin{cases} \lambda _{i} ^{-2 } \ln ^{2/3} \lambda _{i} & \textit{if } n = 6, \\ \lambda _{i}^{ -\min (2,\frac{n-2}{2})} & \textit{if } n \neq 6. \end{cases}\displaystyle \end{aligned}$$

Proof

Taking $U=\sum_{i=1}^{N}\alpha _{i}\delta _{i}$, multiplying $(\mathcal{P}_{\varepsilon})$ by $v_{\varepsilon }$, and integrating on Ω, we obtain

$$\begin{aligned} \int _{\Omega } \vert \nabla v_{\varepsilon } \vert ^{2} + \mu \int _{\Omega }v_{ \varepsilon }^{2}+\mu \sum \alpha _{j} \int _{\Omega }\delta _{j} v_{ \varepsilon }= \int _{\Omega }(U+v_{\varepsilon })^{p+\varepsilon }v_{ \varepsilon }. \end{aligned}$$

(22)

For the right-hand side in (22), we write

$$\begin{aligned} \int _{\Omega }(U+v_{\varepsilon })^{p+\varepsilon }v_{\varepsilon } ={}& \int _{\Omega }U^{p+\varepsilon }v_{\varepsilon }+(p+\varepsilon ) \int _{\Omega }U^{p+\varepsilon -1}v_{\varepsilon }^{2} \\ &{}+O \biggl( \int _{\Omega } \vert v_{\varepsilon } \vert ^{p+1} \biggr) + O_{( n \leq 5) } \biggl( \int _{\Omega }U^{p-2} \vert v_{\varepsilon } \vert ^{3} \biggr) \\ ={}& \int _{\Omega }U^{p+\varepsilon }v_{\varepsilon }+(p+\varepsilon ) \int _{\Omega }U^{p+\varepsilon -1}v_{\varepsilon }^{2} + o \bigl( \Vert v_{ \varepsilon } \Vert ^{2}\bigr), \end{aligned}$$

(23)

where we have used Remark 2.2 and where the notation $O_{( n\leq 5) }$ means that the term appears only if $n\leq 5$.

However, using Lemma 2.1 and (4), we have

$$\begin{aligned} (p+\varepsilon ) \int _{\Omega }U^{p+\varepsilon -1}v_{\varepsilon }^{2}=p \sum_{j=1}^{N} \int _{\Omega }\delta _{j}^{\frac{4}{n-2}}v_{ \varepsilon }^{2} + o\bigl( \Vert v_{\varepsilon } \Vert ^{2}\bigr). \end{aligned}$$

(24)

For the other integral in (23), using Lemma 6.1, we see that, for $n\geq 6$, we have

$$\begin{aligned} \int _{\Omega}U^{p+\varepsilon} v_{\varepsilon }= \sum _{i=1}^{N} \alpha _{i} ^{p+\varepsilon} \int _{\Omega}\delta _{i}^{p+\varepsilon} v_{\varepsilon }+ \sum_{i\neq j} O \biggl( \int _{\Omega}(\delta _{i} \delta _{j})^{\frac{p+\varepsilon}{2}} \vert v_{\varepsilon } \vert \biggr). \end{aligned}$$

(25)

Using Lemmas 2.1, 6.2, and 6.4, we obtain

$$\begin{aligned} \int _{\Omega}(\delta _{i}\delta _{j})^{\frac{p+\varepsilon}{2}} \vert v_{\varepsilon } \vert &\leq c \int _{\Omega}(\delta _{i} \delta _{j})^{\frac{p}{2}} \vert v_{\varepsilon } \vert \\ &\leq c \Vert v_{ \varepsilon } \Vert \biggl( \int _{\Omega}(\delta _{i}\delta _{j})^{ \frac{n}{n-2}} \biggr) ^{\frac{n+2}{2n}}\leq c \Vert v_{\varepsilon } \Vert \bigl[ \varepsilon _{ij}^{\frac{n}{n-2}}\ln \varepsilon _{ij}^{-1} \bigr]^{\frac{n+2}{2n}}. \end{aligned}$$

(26)

For the first term on the right-hand side of (25), using again Lemmas 6.2 and 6.4, we obtain

$$\begin{aligned} \int _{\Omega}\delta _{i}^{p+\varepsilon} v_{\varepsilon }& = c_{0}^{ \varepsilon}\lambda _{i}^{ \varepsilon \frac{n-2}{2}} \int _{\Omega} \delta _{i}^{p} v_{\varepsilon }+O \biggl( \varepsilon \int _{\Omega} \delta _{i}^{p} \vert v_{\varepsilon } \vert \ln \bigl(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2}\bigr) \biggr) \\ & = c_{0}^{\varepsilon}\lambda _{i}^{\varepsilon \frac{n-2}{2}} \int _{ \Omega}\delta _{i}^{p} v_{\varepsilon }+ O \biggl( \varepsilon \Vert v_{ \varepsilon } \Vert \biggl( \int _{\mathbb{R}^{n}}\delta _{i}^{p+1} \ln ^{ \frac{2n}{n+2}}\bigl(1+\lambda _{i} ^{2} \vert x-a_{i} \vert ^{2}\bigr) \biggr)^{\frac{n+2}{2n}} \biggr) \\ & = c_{0}^{\varepsilon}\lambda _{i}^{\varepsilon \frac{(n-2)}{2}} \int _{\Omega}\delta _{i}^{p} v_{\varepsilon }+ O\bigl(\varepsilon \Vert v_{ \varepsilon } \Vert \bigr). \end{aligned}$$

(27)

However, since $v _{\varepsilon }\in E_{a,\lambda }$, using Holder’s inequality and Lemma 6.7, it holds

$$\begin{aligned} \biggl\vert \int _{\Omega}\delta _{i}^{p} v_{\varepsilon } \biggr\vert & = \biggl\vert \int _{\Omega}-\Delta \delta _{i} v_{\varepsilon } \biggr\vert \\ &= \biggl\vert \int _{\partial \Omega } \frac{\partial \delta _{i}}{\partial \nu } v_{ \varepsilon } \biggr\vert \leq \frac{c}{\lambda _{i}^{(n-2)/2}} \int _{ \partial \Omega} \frac{ \vert v_{\varepsilon } \vert }{ \vert x-a_{i} \vert ^{n-1} } \leq \frac{c}{ ( \lambda _{i} d_{i} ) ^{(n-2)/2}} \Vert v_{\varepsilon } \Vert . \end{aligned}$$

(28)

For $n \leq 5$, using Lemma 6.1, we write

$$\begin{aligned} \int _{\Omega} U^{p+\varepsilon} v_{\varepsilon }= \sum _{i=1}^{N} \alpha _{i}^{ p+\varepsilon } \int _{\Omega}\delta _{i}^{p+ \varepsilon} v_{\varepsilon }+ \sum_{i\neq j} O \biggl( \int _{\Omega} \delta _{i}^{p-1+\varepsilon} \delta _{j} \vert v_{\varepsilon } \vert + \int _{\Omega}\delta _{i}\delta _{j}^{p-1+\varepsilon} \vert v_{\varepsilon } \vert \biggr). \end{aligned}$$

(29)

Using Lemmas 6.2 and 6.6, we get (since $1\leq 2n/(n+2) < 8n/(n^{2} - 4 )$ for $n \leq 5$)

$$\begin{aligned} \int _{\Omega} \delta _{i}^{p-1+\varepsilon}\delta _{j} \vert v_{ \varepsilon } \vert & \leq c \int _{\Omega} \delta _{i}^{ p-1 } \delta _{j} \vert v_{\varepsilon } \vert \leq c \Vert v_{ \varepsilon } \Vert \biggl( \int \delta _{i}^{8n/(n^{2} - 4 )} \delta _{j}^{2n/(n+2)} \biggr)^{(n+2)/(2n)} \leq c \Vert v_{\varepsilon } \Vert \varepsilon _{ij}. \end{aligned}$$

(30)

Lastly, by easy computations, we have

$$\begin{aligned} \int _{\Omega }\delta _{j} \vert v_{\varepsilon } \vert \leq C \Vert v_{\varepsilon } \Vert \biggl( \int _{\Omega }\delta _{j}^{ 2n/(n+2)} \biggr)^{(n+2)/(2n)} \leq C \Vert v_{\varepsilon } \Vert T(\lambda _{j}) \end{aligned}$$

(31)

with

$$\begin{aligned} T(\lambda ):= \biggl( \frac{1}{\lambda ^{({n-2})/{2}}} \text{ if } n\leq 5; \frac{ \ln ^{{2}/{3}}(\lambda ) }{ \lambda ^{2}} \text{ if } n=6 ; \frac{ 1 }{\lambda ^{2}} \text{ if } n\geq 7 \biggr). \end{aligned}$$

(32)

Combining Proposition 2.3 and the above estimates, we easily obtain Proposition 2.4. □

3 Estimate of the gradient in the neighborhood of bubbles

As our proof is based on an argument by contradiction, we will assume that problem $(\mathcal{P}_{\varepsilon})$ has a solution $u:= u_{\varepsilon }$ in the form (3) and satisfying (4), and we will need to give careful estimates of some integrals involved in our proof. In fact, for future use, we are going to give some crucial estimates in a more general situation than ours. To this aim, for $N \in \mathbb{N}$, ε and η small positive reals, taking $(\alpha,a,\lambda ) \in \mathcal{V}(N,\varepsilon,\eta )$, where $\mathcal{V}(N,\varepsilon,\eta )$ is defined by (9), and $u=\sum_{i=1}^{N} \alpha _{i}\delta _{a_{i},\lambda _{i}} +v$ with $v\in E_{a,\lambda }$, we need to evaluate the following expressions:

$$\begin{aligned} \int _{\Omega }\nabla u \nabla \psi _{i} + \mu \int _{\Omega }u \psi _{i} - \int _{\Omega } \vert u \vert ^{p-1+\varepsilon }u \psi _{i}\quad \psi _{i}\in \biggl\{ \delta _{i}, \lambda _{i} \frac{\partial \delta _{i}}{\partial \lambda _{i}}, \frac{1}{\lambda _{i}}\frac{\partial \delta _{i}}{\partial a_{i}} \biggr\} ,\quad 1\leq i \leq N. \end{aligned}$$

(33)

We start by dealing with the nonlinear integrals in (33).

Proposition 3.1

Let $n \geq 3$ and $u:= \sum_{i=1}^{N} \alpha _{i} \delta _{a_{i}, \lambda _{i} } + v$ be such that $(\alpha,a,\lambda ) \in \mathcal{V}(N,\varepsilon,\eta )$ and $v \in E_{a,\lambda }$. Then, for $1\leq i\leq N$, we have

$$\begin{aligned} &\int _{\Omega } \vert u \vert ^{p-1+\varepsilon }u \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} \\ &\quad= \overline{c}_{2} \sum _{j\neq i} \alpha _{j} \frac{1}{\lambda _{i}} \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \biggl[ \sum_{k=i,j} \alpha _{k}^{p-1+\varepsilon }\lambda _{k}^{\varepsilon \frac{ n-2}{2}} \biggr]+ O \biggl( R_{3,i} + \sum_{k\neq r}\varepsilon _{kr}^{ \frac{n}{n-2}}\ln \varepsilon _{kr}^{-1} + \varepsilon _{kr} ^{2} \biggr), \end{aligned}$$

where

$$\begin{aligned} R_{3,i}:= {}& \sum_{j\neq i} \bigl[ \lambda _{j} \vert a_{i}-a_{j} \vert \varepsilon _{ij}^{\frac{n+1}{n-2}} + \varepsilon \varepsilon _{ij} \bigr] + \frac{1}{ ( \lambda _{i} d_{i} )^{n /2}} \sum_{k=1}^{N} \frac{1}{ (\lambda _{k} d_{k} ) ^{(n+2)/2}} \\ &{} + \Vert v \Vert \biggl( \Vert v \Vert + \varepsilon + \frac{1}{ ( \lambda _{i} d_{i} ) ^{n/2} } + \sum_{j\neq i} \bigl[ \varepsilon _{ij} + \varepsilon _{ij}^{(n+2)/(2(n-2))} \ln \bigl( \varepsilon _{ij}^{-1}\bigr)^{(n+2)/(2n)} \bigr] \biggr). \end{aligned}$$

Proof

To simplify notation, we write $U=\sum_{j=1}^{N}\alpha _{j}\delta _{j}$ and $\psi _{i}=\frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}}$. We set

$$\begin{aligned} \Omega _{v}=\bigl\{ x\in \Omega: \bigl\vert v(x) \bigr\vert \leq U(x)\bigr\} \quad\text{and}\quad \Omega _{v}^{c}=\Omega \setminus \Omega _{v}. \end{aligned}$$

Using Lemma 6.1, we get

$$\begin{aligned} &\int _{\Omega } \vert u \vert ^{p-1+\varepsilon }u\psi _{i} \\ &\quad= \int _{\Omega }U^{p+ \varepsilon }\psi _{i} +(p+ \varepsilon ) \int _{\Omega }U^{p-1+ \varepsilon }v\psi _{i} +O \biggl( \int _{\Omega _{v}}U^{p-2+\varepsilon }v^{2} \vert \psi _{i} \vert + \int _{\Omega _{v}^{c}} \vert v \vert ^{p+\varepsilon } \vert \psi _{i} \vert \biggr). \end{aligned}$$

(34)

First, from Lemmas 6.3 and 6.2, we obtain

$$\begin{aligned} \int _{\Omega _{v}}U^{p-2+\varepsilon }v^{2} \vert \psi _{i} \vert + \int _{ \Omega _{v}^{c}} \vert v \vert ^{p+\varepsilon } \vert \psi _{i} \vert \leq c \int _{ \Omega }U^{p-1}v^{2} + c \int _{\Omega } \vert v \vert ^{p+1} \leq c \Vert v \Vert ^{2}. \end{aligned}$$

(35)

Second, Lemmas 6.1 and 6.2 imply that

$$\begin{aligned} \int _{\Omega }U^{p-1+\varepsilon }v\psi _{i} ={} & \int _{\Omega }( \alpha _{i}\delta _{i})^{p-1+\varepsilon }v \psi _{i} + O \biggl( \int _{ \Omega _{i}}(\alpha _{i}\delta _{i})^{p-2} \biggl(\sum_{j\neq i}\alpha _{j} \delta _{j}\biggr) \vert v \vert \vert \psi _{i} \vert \biggr) \\ &{}+O \biggl( \int _{\Omega _{i}^{c}}\biggl(\sum_{j\neq i}\alpha _{j}\delta _{j}\biggr)^{p-1} \vert v \vert \vert \psi _{i} \vert \biggr), \end{aligned}$$

(36)

where $\Omega _{i}=\{x\in \Omega: \sum_{j\neq i}\alpha _{j}\delta _{j}(x) \leq \alpha _{i}\delta _{i}(x)\}$ and $\Omega _{i}^{c}= \Omega \setminus \Omega _{i}$.

To deal with the remaining term in (36), we distinguish two cases. For $n\geq 6$, observe that $p-1:= 4/(n-2) \leq 1$. Thus, using Lemmas 6.3 and 6.4, it follows that

$$\begin{aligned} &\int _{\Omega _{i}}(\alpha _{i}\delta _{i})^{p-2} \biggl(\sum_{j\neq i} \alpha _{j}\delta _{j}\biggr) \vert v \vert \vert \psi _{i} \vert + \int _{\Omega _{i}^{c}}\biggl( \sum_{j\neq i} \alpha _{j}\delta _{j}\biggr)^{p-1} \vert v \vert \vert \psi _{i} \vert \\ &\quad\leq C \int _{\Omega _{i}}(\alpha _{i}\delta _{i})^{p-1} \biggl(\sum_{j \neq i}\alpha _{j}\delta _{j}\biggr) \vert v \vert + C \int _{\Omega _{i}^{c}}\biggl( \sum_{j\neq i} \alpha _{j}\delta _{j}\biggr)^{p-1} \vert v \vert \delta _{i} \\ &\quad\leq C \sum_{j\neq i} \int _{\Omega }(\delta _{i}\delta _{j})^{{p}/{2}} \vert v \vert \leq c \Vert v \Vert \biggl( \int _{\Omega }(\delta _{i}\delta _{j})^{{n}/{(n-2)}} \biggr)^{{(n+2)}/{(2n)}} \\ &\quad\leq C \Vert v \Vert \sum \varepsilon _{ij}^{{(n+2)}/{(2(n-2))}} \ln \bigl( \varepsilon _{ij}^{-1} \bigr) ^{{(n+2)}/{(2n)}}. \end{aligned}$$

(37)

For $n\leq 5$, we have $p-1>1$, and therefore, using (30), it holds

$$\begin{aligned} &\int _{\Omega _{i}}(\alpha _{i}\delta _{i})^{p-2} \biggl(\sum_{j\neq i} \alpha _{j}\delta _{j}\biggr) \vert v \vert \vert \psi _{i} \vert + \int _{\Omega _{i}^{c}}\biggl( \sum_{j\neq i} \alpha _{j}\delta _{j}\biggr)^{p-1} \vert v \vert \vert \psi _{i} \vert \\ &\quad\leq c \sum _{j\neq i} \biggl( \int _{\Omega }\delta _{i}^{p-1} \delta _{j} \vert v \vert + \int _{\Omega }\delta _{j}^{p-1} \vert v \vert \delta _{i} \biggr) \\ & \quad\leq c \Vert v \Vert \sum \varepsilon _{ij}. \end{aligned}$$

(38)

To complete estimate (36), using Lemmas 6.2. 6.3, 6.4, 6.7 and $v \in E_{a,\lambda }$, we obtain

$$\begin{aligned} \int _{\Omega }\delta _{i}^{p-1+\varepsilon }v\psi _{i}&=c_{0}^{ \varepsilon }\lambda _{i}^{{\varepsilon (n-2)}/{2}} \int _{\Omega } \delta _{i} ^{p-1}v\psi _{i} +O \biggl(\varepsilon \int _{\Omega } \delta _{i}^{p-1} \vert v \vert \vert \psi _{i} \vert \ln \bigl(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2} \bigr) \biggr) \\ &=c_{0}^{\varepsilon }\lambda _{i}^{{\varepsilon (n-2)}/{2}} \biggl[ \frac{1}{p} \langle \nabla \psi _{i}, \nabla v \rangle _{L^{2}} - \int _{\partial \Omega }\frac{\partial \psi _{i}}{\partial \nu} v \biggr] +O\bigl(\varepsilon \Vert v \Vert \bigr) \end{aligned}$$

(39)

$$\begin{aligned} & = O \biggl( \frac{1}{ \lambda _{i} ^{n/2} } \int _{ \partial \Omega } \frac{ \vert v \vert }{ \vert x - a _{i} \vert ^{n } } + \varepsilon \Vert v \Vert \biggr) \\ &=O \biggl( \Vert v \Vert \biggl[\varepsilon + \frac{ 1}{ ( \lambda _{i} d_{i} )^{n/2}} \biggr] \biggr). \end{aligned}$$

(40)

It remains to estimate the first integral on the right-hand side of (34). To this aim, by Lemma 6.1, we write

$$\begin{aligned} \int _{\Omega }U^{p+\varepsilon } \psi _{i} = {}& \int _{\Omega }( \alpha _{i}\delta _{i})^{p+\varepsilon } \psi _{i} +(p+\varepsilon ) \int _{\Omega }(\alpha _{i}\delta _{i})^{p+\varepsilon -1} \biggl(\sum_{j \neq i}\alpha _{j}\delta _{j}\biggr) \psi _{i} + \int _{\Omega }\biggl(\sum_{j\neq i}\alpha _{j}\delta _{j}\biggr)^{p+ \varepsilon } \psi _{i} \\ &{}+ O \biggl( \int _{\Omega _{i}} \biggl[(\alpha _{i} \delta _{i})^{p-2+\varepsilon } \biggl(\sum_{j\neq i} \alpha _{j}\delta _{j}\biggr)^{2} + \biggl(\sum _{j\neq i}\alpha _{j}\delta _{j} \biggr)^{p+\varepsilon } \biggr] \vert \psi _{i} \vert \biggr) \\ &{}+O \biggl( \int _{\Omega _{i}^{c}} \biggl[\alpha _{i}\delta _{i} \biggl(\sum_{j \neq i}\alpha _{j}\delta _{j}\biggr)^{p+\varepsilon -1} + (\alpha _{i} \delta _{i})^{p-1+\varepsilon } \biggl(\sum _{j\neq i}\alpha _{j}\delta _{j}\biggr) \biggr] \vert \psi _{i} \vert \biggr). \end{aligned}$$

(41)

We are going to estimate each term of (41). First, for $n \geq 4$, it follows that $p-1 \leq 2$. Using Lemmas 6.2, 6.3, and 6.4, we obtain

$$\begin{aligned} \int _{\Omega _{i}} [ \cdots ] \vert \psi _{i} \vert + \int _{ \Omega _{i}^{c}} [ \cdots ] \vert \psi _{i} \vert \leq c \sum_{j \neq i} \int _{\Omega }(\delta _{i}\delta _{j})^{\frac{n}{n-2}} \leq c \sum_{j\neq i}\varepsilon _{ij}^{\frac{n}{n-2}} \ln \varepsilon _{ij}^{-1}, \end{aligned}$$

(42)

and for $n = 3$, using Lemmas 6.2, 6.3, and 6.6, it follows that

$$\begin{aligned} \int _{\Omega _{i}} [ \cdots ] \vert \psi _{i} \vert + \int _{ \Omega _{i}^{c}} [ \cdots ] \vert \psi _{i} \vert \leq c \sum_{j \neq i} \int _{\Omega }\delta _{i} ^{4} \delta _{j} ^{2} + \delta _{i} ^{2} \delta _{j} ^{4} \leq c \sum_{j\neq i} \varepsilon _{ij}^{2}. \end{aligned}$$

Second, since $\psi _{i}$ is odd and $\delta _{i}$ is even with respect to $x-a_{i}$, using Lemmas 6.2, 6.3, and 6.4, we get

$$\begin{aligned} \int _{\Omega }(\alpha _{i}\delta _{i})^{p+\varepsilon } \psi _{i} & = O \biggl( \int _{\Omega \setminus B(a_{i}, d_{i} )} \delta _{i}^{p} \vert \psi _{i} \vert \biggr) \\ &= O \biggl( \int _{\Omega \setminus B(a_{i},d_{i} )} \frac{\delta _{i}^{p+1}}{\lambda _{i} \vert x - a_{i} \vert } \biggr)= O \biggl( \frac{1}{ ( \lambda _{i} d_{i} ) ^{n+1}} \biggr), \end{aligned}$$

(43)

where $d_{i}:= d(a_{i}, \partial \Omega )$.

Third, using Lemmas 6.2, 6.3, 6.4, and 6.6, we obtain

$$\begin{aligned} &\int _{\Omega } \delta _{i} ^{p+\varepsilon -1} \psi _{i} \delta _{j} \\ &\quad= c_{0}^{\varepsilon }\lambda _{i}^{\frac{\varepsilon (n-2)}{2}} \int _{\Omega }\delta _{i}^{p-1} \psi _{i} \delta _{j} + O \biggl( \varepsilon \int _{\Omega }\delta _{i}^{p} \delta _{j}\ln \bigl(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2}\bigr) \biggr) \\ &\quad = c_{0}^{\varepsilon }\lambda _{i}^{\frac{\varepsilon (n-2)}{2}} \int _{\mathbb{R}^{n}} \delta _{i}^{p-1} \psi _{i} \delta _{j} \\ &\qquad{}+ O \biggl( \int _{\mathbb{R}^{n}\setminus \Omega } \frac{\delta _{i}^{p} \delta _{j} }{ \lambda _{i} \vert x - a_{i} \vert }+ \varepsilon \int _{\Omega }\bigl(\delta _{i}^{ \frac{n}{n-2}} \delta _{j} \bigr) \delta _{i}^{\frac{2}{n-2}} \ln \bigl(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2} \bigr) \biggr) \\ &\quad=c_{0}^{\varepsilon }\lambda _{i}^{\frac{\varepsilon (n-2)}{2}} \frac{\overline{c}_{2}}{p\lambda _{i}} \frac{\partial \varepsilon _{ij}}{\partial a_{i}}+O \biggl(\lambda _{j} \vert a_{i}-a_{j} \vert \varepsilon _{ij}^{\frac{n+1}{n-2}} + \frac{1}{ ( \lambda _{i} d_{i} ) ^{{(n+4)}/{2}} } \frac{1}{ ( \lambda _{j} d_{j} ) ^{(n - 2 )/2}}+\varepsilon \varepsilon _{ij} \biggr). \end{aligned}$$

(44)

Finally, using Lemmas 6.1, 6.2, and 6.3, we get

$$\begin{aligned} &\int _{\Omega }\biggl(\sum_{j\neq i}\alpha _{j}\delta _{j}\biggr)^{p+ \varepsilon } \psi _{i} \\ &\quad= \sum_{j\neq i}\alpha _{j}^{p+\varepsilon } \int _{\Omega }\delta _{j} ^{p+\varepsilon }\psi _{i} + O \biggl(\sum_{k \notin \{i,j\}} \int _{\Omega }\delta _{k}\delta _{j}^{p-1} \vert \psi _{i} \vert \biggr) \\ &\quad = \sum_{j\neq i}\alpha _{j}^{p+\varepsilon } c_{0}^{\varepsilon } \lambda _{j}^{\frac{\varepsilon (n-2)}{2}} \int _{\Omega }\delta _{j}^{p} \psi _{i} \\ &\qquad{} +O \biggl(\varepsilon \int _{\Omega }\delta _{j}^{p} \delta _{i} \ln \bigl(1+\lambda _{j}^{2} \vert x-a_{j} \vert ^{2}\bigr) + \sum _{k\notin \{i,j\}} \int _{\Omega }\delta _{k}\delta _{j}^{p-1} \vert \psi _{i} \vert \biggr). \end{aligned}$$

(45)

On the other hand, using Lemmas 6.3, 6.4, and 6.6 we see that

$$\begin{aligned} & \sum_{k\notin \{i,j\}} \int _{\Omega }\delta _{k}\delta _{j}^{p-1} \vert \psi _{i} \vert \leq \sum_{l\neq r} \int _{\Omega }(\delta _{l}\delta _{r})^{n/(n-2)} \leq c \sum_{l\neq r}\varepsilon _{lr}^{n/(n-2)} \ln \varepsilon _{lr}^{-1}, \quad\text{if } n \geq 4, \end{aligned}$$

(46)

$$\begin{aligned} & \sum_{k\notin \{i,j\}} \int _{\Omega }\delta _{k}\delta _{j}^{p-1} \vert \psi _{i} \vert \leq \sum_{r\neq j} \int _{\Omega }\delta _{j} ^{4} \delta _{r}^{2} \leq c \sum_{l\neq r} \varepsilon _{lr}^{2}, \quad\text{if } n = 3, \\ & \int _{\Omega }\delta _{j}^{p} \vert \delta _{i} \vert \ln \bigl(1+\lambda _{j}^{2} \vert x-a_{j} \vert ^{2}\bigr) \leq \int _{\Omega }\bigl(\delta _{i}\delta _{j} ^{n/(n-2)}\bigr) \delta _{j}^{2/(n-2) } \ln \bigl(1+\lambda _{j}^{2} \vert x-a_{j} \vert ^{2} \bigr) \leq c \varepsilon _{ij}, \end{aligned}$$

(47)

$$\begin{aligned} & \int _{\Omega }\delta _{j}^{p}\psi _{i} = \int _{\mathbb{R}^{n}} \delta _{j}^{p}\psi _{i} + O \biggl( \int _{\mathbb{R}^{n}\setminus \Omega } \frac{\delta _{j}^{p}\delta _{i}}{\lambda _{i} \vert x - a_{i} \vert } \biggr) \\ &\phantom{\int _{\Omega }\delta _{j}^{p}\psi _{i} } = \frac{\overline{c}_{2}}{\lambda _{i}} \frac{\partial \varepsilon _{ij}}{\partial a_{i}} + O \biggl(\lambda _{j} \vert a_{i}-a_{j} \vert \varepsilon _{ij}^{\frac{n+1}{n-2}} + \frac{1}{ ( \lambda _{j} d_{j} )^{\frac{(n+2)}{2}}} \frac{1}{ ( \lambda _{i} d_{i} )^{\frac{n}{2}}} \biggr). \end{aligned}$$

(48)

Combining the above estimates, the proof of Proposition 3.1 follows. □

Next, we are going to improve Proposition 3.1 in some particular cases. More precisely, we need to improve the term $\sum \varepsilon _{kr}^{n/(n-2)}\ln \varepsilon _{kr}^{-1}$ in these cases.

Proposition 3.2

Let $n \geq 5$, and for $1\leq i\leq N$, let $N_{i}:=\{j: 1\leq j\leq N, |a_{i}-a_{j}|\to 0\}$, $\gamma:= \min \{|a_{j}-a_{k}|, j\neq k, j, k \in N_{i}\}$, and $\sigma:= \max \{|a_{j}-a_{k}|, j\neq k, j, k \in N_{i}\}$. Assume that ${\gamma} / {\sigma} \geq c >0$ and $\gamma \min \{\lambda _{k}: k\in N_{i}\} \geq c>0$. Then, for $1\leq i\leq N$, we have

$$\begin{aligned} \int _{\Omega } \vert u \vert ^{p-1+\varepsilon }u \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} = \overline{c}_{2} \sum _{j\neq i} \alpha _{j} \frac{1}{\lambda _{i}} \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \bigl[\alpha _{j}^{p-1+ \varepsilon }\lambda _{j}^{\frac{\varepsilon (n-2)}{2}} + \alpha _{i}^{p-1+ \varepsilon }\lambda _{i}^{\frac{\varepsilon (n-2)}{2}} \bigr] + O( R_{3,i} + R_{4,i} ), \end{aligned}$$

where $R_{3,i}$ is defined in Proposition 3.1and

$$\begin{aligned} R_{4,i} = \frac{1}{\gamma \lambda _{i}} \sum_{k\neq r} \varepsilon _{kr}^{ \frac{n}{n-2}} +\sum_{j\neq i} \varepsilon _{ij}^{ \frac{n+{1}/{2}}{n-2}}+ \frac{1}{\gamma \lambda _{i}} \sum _{j\neq i} \varepsilon _{ij}^{\frac{n-{1}/{2}}{n-2}}. \end{aligned}$$

Proof

The proof follows the proof of Proposition 3.1, but we need to improve estimates (42) and (46). We remark that, since the distances $| a_{k} - a_{j} |$s are of the same order (that is $\gamma / \sigma \geq c > 0$) and $\gamma \min \{ \lambda _{k}: k \in N_{i} \} \geq c$ (by the assumption of the proposition), it follows from Assertion (4) of Lemma 6.5 that $\varepsilon _{kj}$ and $(\lambda _{k} \lambda _{j} \gamma ^{2})^{(2-n)/2}$ are of the same order.

We start by improving (46). Let $B_{\ell }=B(a_{\ell},\gamma /4)$, we write for $k\notin \{i,j\}$ and $i\neq j$

$$\begin{aligned} \int _{\Omega }\delta _{k}\delta _{j}^{p-1} \vert \psi _{i} \vert = \int _{B_{i}} \cdots + \int _{B_{j}} \cdots + \int _{B_{k}} \cdots + \int _{\Omega \setminus (B_{i}\cup B_{i}\cup B_{k})} \cdots:= (I) + (\mathit{II}) + (\mathit{III}) + (\mathit{IV}). \end{aligned}$$

For the last one, using Lemma 6.3, it holds

$$\begin{aligned} (\mathit{IV}) &\leq \frac{C}{\lambda _{k}^{(n-2)/2}\lambda _{i}^{n/2} \lambda _{j}^{2}} \int _{\Omega \setminus (B_{i}\cup B_{i}\cup B_{k})} \frac{dx}{ \vert x-a_{k} \vert ^{n-2} \vert x-a_{i} \vert ^{n-1} \vert x-a_{j} \vert ^{4}} \\ &\leq \frac{C}{\lambda _{k}^{(n-2)/2}\lambda _{i}^{n/2} \lambda _{j}^{2}\gamma ^{n+1}} \leq \frac{C}{\lambda _{i}\gamma}\sum _{l\neq r}\varepsilon _{lr}^{n/(n-2)}. \end{aligned}$$

Concerning the first one, it holds

$$\begin{aligned} (I) &\leq \frac{C}{(\lambda _{k}\gamma ^{2})^{\frac{n-2}{2}}(\lambda _{j}\gamma ^{2})^{2}} \int _{B_{i}}\frac{dx}{\lambda _{i}^{\frac{n}{2}} \vert x-a_{i} \vert ^{n-1}} \leq \frac{C}{\lambda _{k}^{\frac{n-2}{2}}\lambda _{i}^{\frac{n}{2}} \lambda _{j}^{2}\gamma ^{n+1}} \leq \frac{C}{\lambda _{i}\gamma}\sum_{l\neq r}\varepsilon _{lr}^{ \frac{n}{n-2}}, \end{aligned}$$

and in the same way we obtain

$$\begin{aligned} (\mathit{II}) + (\mathit{III}) \leq \frac{C}{\lambda _{i}\gamma}\sum_{l\neq r} \varepsilon _{lr}^{\frac{n}{n-2}}. \end{aligned}$$

This completes the desired improvement for (46).

Now, we will focus on the improvement for (42). Using Lemmas 6.2 and 6.3, we get

$$\begin{aligned} &\int _{\Omega _{i}}[ \cdots ] \vert \psi _{i} \vert + \int _{ \Omega _{i}^{c}} [ \cdots ] \vert \psi _{i} \vert \\ &\quad\leq c \sum_{j \neq i} \int _{\Omega }\delta _{j}^{\frac{n+{1}/{2}}{n-2}} \delta _{i}^{ \frac{n-{1}/{2}}{n-2}} \frac{1}{\lambda _{i} \vert x-a_{i} \vert } \\ &\quad\leq \sum_{j\neq i} \frac{c }{\gamma \lambda _{i}} \int _{\Omega \setminus B_{i}} \delta _{j}^{\frac{n+{1}/{2}}{n-2}} \delta _{i}^{ \frac{n-{1}/{2}}{n-2}} +c\sum_{j\neq i} \frac{c }{(\gamma ^{2} \lambda _{j})^{\frac{n+1/2}{2}}} \int _{B_{i}} \frac{1}{\lambda _{i} \vert x-a_{i} \vert } \frac{\lambda _{i}^{\frac{n-1/2}{n-2}}}{(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2})^{\frac{n-1/2}{n-2}}} \\ &\quad \leq \sum_{j\neq i}\frac{c }{\gamma \lambda _{i}} \varepsilon _{ij}^{ \frac{n-1/2}{n-2}} + c\sum_{j\neq i} \frac{c }{(\gamma ^{2} \lambda _{j})^{\frac{n+1/2}{2}}} \frac{1}{\lambda _{i}^{\frac{n+1/2}{n-2}}} \\ & \quad\leq \sum_{j\neq i} \frac{c }{\gamma \lambda _{i}} \varepsilon _{ij}^{ \frac{n-1/2}{n-2}} + c \varepsilon _{ij}^{\frac{n+1/2}{n-2}}, \end{aligned}$$

where we have used Lemma 6.6. This completes the improvement of (42). Hence the proof of Proposition 3.2 follows. □

Next, we are going to deal with the linear terms in (33). We start by the second one, namely, we prove the following.

Proposition 3.3

Let $n \geq 3$ and $u:= \sum_{i=1}^{N} \alpha _{i} \delta _{a_{i}, \lambda _{i} } + v$ be such that $(\alpha,a,\lambda ) \in \mathcal{V}(N,\varepsilon,\eta )$. Then, for $1\leq i\leq N$, the following fact holds:

$$\begin{aligned} \biggl\vert \int _{\Omega }u \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} \biggr\vert \leq c R_{5,i}\quad \textit{with } R_{5,i}:= \frac{ 1 }{\lambda _{i}} \sum_{j \ne i} \varepsilon _{ij} + \textstyle\begin{cases} \frac{ \vert \ln d_{i} \vert }{\lambda _{i} ^{2} } + \Vert v \Vert \frac{ 1 }{\lambda _{i} ^{3/2} } & \textit{if } n = 3, \\ \frac{ 1 }{\lambda _{i} ^{3} d_{i} } + \Vert v \Vert \frac{ (\ln \lambda _{i})^{3/4} }{\lambda _{i} ^{2} } & \textit{if } n = 4, \\ \frac{ 1 }{\lambda _{i} ^{2} ( \lambda _{i} d_{i} )^{n-3} } + \Vert v \Vert \frac{ 1 }{\lambda _{i} ^{2} } & \textit{if } n \geq 5. \end{cases}\displaystyle \end{aligned}$$

Proof

Let $\psi _{i}:= \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}}$. Observe that, for $j \neq i$, using Lemmas 6.3 and 6.6, it holds

$$\begin{aligned} \int _{\Omega }\delta _{j} \vert \psi _{i} \vert \leq c \int _{\Omega } \frac{1}{\lambda _{i} \vert x - a _{i} \vert }\delta _{j} \delta _{i} \leq \frac{ c }{\lambda _{i} } \biggl( \int _{\Omega }(\delta _{j} \delta _{i})^{ \frac{n-1}{n-2}} \biggr)^{\frac{n-2}{n-1}} \biggl( \int _{\Omega } \frac{1}{ \vert x - a_{i} \vert ^{n-1}} \biggr)^{\frac{ 1 }{n-1}} \leq \frac{ c }{\lambda _{i} } \varepsilon _{ij}. \end{aligned}$$

For $j = i$, let $R>0$ be such that $\Omega \subset B(a_{i}, R)$, it holds that

$$\begin{aligned} \biggl\vert \int _{\Omega }\delta _{i} \psi _{i} \biggr\vert & = \biggl\vert \int _{B(a_{i}, d_{i})} \delta _{i} \psi _{i} + \int _{\Omega \setminus B(a_{i}, d_{i})} \delta _{i} \psi _{i} \biggr\vert \leq c \int _{\Omega \setminus B(a_{i}, d_{i})} \frac{1}{\lambda _{i} \vert x - a _{i} \vert }\delta _{i} ^{2} \\ & \leq \frac{ c }{ \lambda _{i} ^{2} } \int _{ \lambda _{i} d_{i} } ^{ \lambda _{i} R } \frac{ t^{n-2} }{ ( 1 + t ^{2} )^{n-2} } \leq c \biggl( \frac{ \vert \ln d_{i} \vert }{\lambda _{i} ^{2} } \text{ if } n =3; \frac{1}{\lambda _{i} ^{2} (\lambda _{i} d_{i} ) ^{n-3} } \text{ if } n \geq 4 \biggr). \end{aligned}$$

Finally, using again Lemma 6.3, it holds

$$\begin{aligned} \int _{\Omega } \vert v \vert \vert \psi _{i} \vert &\leq c \Vert v \Vert \biggl( \int _{ \Omega } \biggl( \frac{1}{\lambda _{i} \vert x - a _{i} \vert }\delta _{i} \biggr)^{ \frac{2n}{n+2}} \biggr)^{\frac{n+2}{2n}} \\ & \leq c \| v \| \frac{1}{\lambda _{i} ^{2} } \biggl( \int _{0}^{ \lambda _{i} R } \frac{ t^{n-1-\frac{2n}{n+2}} }{ (1+ t^{2} ) ^{ n(n-2)/(n+2)}} \biggr) ^{ \frac{n+2}{2n}} \leq c \Vert v \Vert \times \textstyle\begin{cases} 1 / \lambda _{i} ^{ 3 / 2 } & \text{if } n = 3, \\ (\ln \lambda _{i})^{3/4} / \lambda _{i} ^{ 2 } & \text{if } n = 4, \\ 1 / \lambda _{i} ^{2} & \text{if } n \geq 5. \end{cases}\displaystyle \end{aligned}$$

Thus the proof follows. □

Next, we deal with the first linear term in (33).

Proposition 3.4

Let $n \geq 3$ and $u:= \sum_{i=1}^{N} \alpha _{i} \delta _{a_{i}, \lambda _{i} } + v$ be such that $(\alpha,a,\lambda ) \in \mathcal{V}(N,\varepsilon,\eta )$ and $v \in E_{a,\lambda }$. Then, for $1\leq i\leq N$, the following fact holds:

$$\begin{aligned} & \int _{\Omega }\nabla u \cdot \nabla \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} = \overline{c}_{2} \frac{1}{\lambda _{i}} \sum _{j \neq i } \alpha _{j} \frac{ \partial \varepsilon _{ij}}{\partial a_{i} } + O ( R_{6,i} ) \quad\textit{with } \\ & R_{6,i}:= \frac{ 1}{ ( \lambda _{i} d_{i}) ^{\frac{ n}{2 }}} \sum_{k} \frac{ 1}{ ( \lambda _{k} d_{k}) ^{\frac{ n-2}{2 }}} + \sum_{j \neq i } \lambda _{j} \vert a_{i} - a_{j} \vert \varepsilon _{ij}^{\frac{ n+1 }{n-2}}. \end{aligned}$$

Proof

Since $v \in E_{a,\lambda }$, it follows that

$$\begin{aligned} &\int _{\Omega }\nabla u \cdot \nabla \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} \\ &\quad = \sum_{1 \leq j \leq N} \alpha _{j} \int _{\Omega }\nabla \delta _{j} \cdot \nabla \frac{1}{\lambda _{i}}\frac{\partial \delta _{i}}{\partial a_{i}} \\ &\quad = \sum_{1 \leq j \leq N} \alpha _{j} \int _{\mathbb{R}^{n}} \nabla \delta _{j} \nabla \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} + O \biggl[ \biggl( \int _{ \mathbb{R}^{n} \setminus \Omega } \vert \nabla \delta _{j} \vert ^{2} \biggr)^{1/2} \biggl( \int _{\mathbb{R}^{n} \setminus \Omega } \biggl\vert \nabla \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} \biggr\vert ^{2} \biggr)^{1/2} \biggr] \\ &\quad = \sum_{j \neq i} \alpha _{j} \biggl( \overline{c}_{2} \frac{1}{\lambda _{i}} \frac{ \partial \varepsilon _{ij}}{\partial a_{i} } + O \bigl( \lambda _{j} \vert a_{i} - a_{j} \vert \varepsilon _{ij}^{\frac{ n+1 }{n-2}} \bigr) \biggr) + O \biggl( \frac{ 1}{ ( \lambda _{i} d_{i} ) ^{\frac{ n}{2 }}} \sum_{j} \frac{ 1}{ ( \lambda _{j} d_{j} ) ^{\frac{ n-2}{2 }}} \biggr), \end{aligned}$$

where we have used Lemma 6.4. This completes the proof. □

Combining Propositions 3.1, 3.3, and 3.4, we obtain the following balancing expression involving the point of concentration $a_{i}$.

Proposition 3.5

Let $n \geq 3$ and $u = \sum_{ j \leq N} \alpha _{j} \delta _{a_{j},\lambda _{j}} + v$ be such that $(\alpha,a,\lambda ) \in \mathcal{V}(N,\varepsilon,\eta )$ and $v \in E_{a,\lambda }$. Then, for $1\leq i\leq N$, the following fact holds:

$$\begin{aligned} &\int _{\Omega }\nabla u \cdot \nabla \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} + \mu \int _{\Omega }u \frac{1}{\lambda _{i}}\frac{\partial \delta _{i}}{\partial a_{i}} - \int _{\Omega } \vert u \vert ^{p-1+\varepsilon }u \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} \\ &\quad = \overline{c}_{2} \sum_{j\neq i} \alpha _{j} \frac{1}{\lambda _{i}} \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \bigl[ 1 - \alpha _{j}^{p-1+ \varepsilon }\lambda _{j}^{\frac{\varepsilon (n-2)}{2}} - \alpha _{i}^{p-1+ \varepsilon }\lambda _{i}^{\frac{\varepsilon (n-2)}{2}} \bigr] + O( R_{35,i} ) \\ & \quad\textit{where } R_{35,i}:= R_{3,i} + R_{5,i} + R_{6,i} + \sum_{k \neq r } \bigl( \varepsilon _{kr}^{\frac{n}{n-2}} \ln \bigl(\varepsilon _{kr}^{-1} \bigr) + \varepsilon _{kr}^{2} \bigr), \end{aligned}$$

and where $\overline{c}_{2}$ is defined in Lemma 6.4and $R_{3,i}$, $R_{5,i}$, $R_{6,i}$ are defined in Propositions 3.1, 3.3, 3.4, respectively.

When the concentration points satisfy some properties, we can improve the previous proposition. More precisely, combining Propositions 3.2, 3.3, and 3.4, we obtain the following.

Proposition 3.6

Let $n \geq 5$ and $u = \sum_{ j \leq N} \alpha _{j} \delta _{a_{j},\lambda _{j}} + v$ be such that $(\alpha,a,\lambda ) \in \mathcal{V}(N,\varepsilon,\eta )$ and $v \in E_{a,\lambda }$. For $i \leq N$, let $N_{i}:=\{j: |a_{i}-a_{j}|\to 0\}$, $\gamma:= \min \{|a_{j}-a_{k}|, j\neq k, j, k \in N_{i}\}$ and $\sigma:= \max \{|a_{j}-a_{k}|, j\neq k, j, k \in N_{i}\}$. Assume that ${\gamma} / {\sigma} \geq c >0$ and $\gamma \min \{\lambda _{k}: k\in N_{i}\} \geq c>0$. Then, for $1\leq i\leq N$, the following fact holds:

$$\begin{aligned} & \int _{\Omega }\nabla u \cdot \nabla \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} + \mu \int _{\Omega }u \frac{1}{\lambda _{i}}\frac{\partial \delta _{i}}{\partial a_{i}} - \int _{\Omega } \vert u \vert ^{p-1+\varepsilon }u \frac{1}{\lambda _{i}} \frac{\partial \delta _{i}}{\partial a_{i}} \\ &\quad= \overline{c}_{2} \sum_{j\neq i} \alpha _{j} \frac{1}{\lambda _{i}} \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \bigl[ 1 - \alpha _{j}^{p-1+ \varepsilon }\lambda _{j}^{\frac{\varepsilon (n-2)}{2}} - \alpha _{i}^{p-1+ \varepsilon }\lambda _{i}^{\frac{\varepsilon (n-2)}{2}} \bigr] + O( R_{3,i} + R_{4,i} + R_{5,i} + R_{6,i} ), \end{aligned}$$

where $\overline{c}_{2}$ is defined in Lemma 6.4and $R_{3,i}$, $R_{4,i}$, $R_{5,i}$, $R_{6,i}$ are defined in Propositions 3.1, 3.2, 3.3, 3.4, respectively.

In the same way, we prove the following balancing expression involving the rate of the concentration and the mutual interaction of bubbles $\varepsilon _{ij}$.

Proposition 3.7

Let $n \geq 4$ and $u = \sum_{ j \leq N} \alpha _{j} \delta _{a_{j},\lambda _{j}} + v$ be such that $(\alpha,a,\lambda ) \in \mathcal{V}(N,\varepsilon,\eta )$ (with $d_{i}:= d(a_{i}, \partial \Omega ) \geq c$ if $n=4$) and $v \in E_{a,\lambda }$. Then, for $1\leq i\leq N$, the following fact holds:

$$\begin{aligned} &\int _{\Omega }\nabla u \cdot \nabla {\lambda _{i}} \frac{\partial \delta _{i}}{\partial \lambda _{i}} + \mu \int _{ \Omega }u {\lambda _{i}} \frac{\partial \delta _{i}}{\partial \lambda _{i}} - \int _{\Omega } \vert u \vert ^{p-1+ \varepsilon } u {\lambda _{i}} \frac{\partial \delta _{i}}{\partial \lambda _{i}} \\ &\quad= - c_{1} \varepsilon - c(n) \mu \frac{ \ln ^{\sigma _{n}} (\lambda _{i}) }{ \lambda _{i} ^{2}} - \overline{c}_{2} \sum _{ j \neq i } \lambda _{i} \frac{ \partial \varepsilon _{ij} }{ \partial \lambda _{i} } \\ &\qquad{} + o \biggl(\varepsilon + \sum_{ k \leq N} \frac{ \ln ^{\sigma _{n}} (\lambda _{k}) }{\lambda _{k} ^{2}} + \sum_{ k\neq r } \varepsilon _{kr} \biggr) + O \biggl( \frac{ 1 }{ ( \lambda _{i} d_{i} )^{(n-2)/2} } \sum \frac{1}{ (\lambda _{k} d_{k} ) ^{(n-2) / 2} } + \Vert v \Vert ^{2} \biggr), \end{aligned}$$

where $c_{1}$ is defined in (49), $c(n)$ is defined in (52) if $n \geq 5$ and in (53) if $n =4$ and $\sigma _{4} = 1$ and $\sigma _{n} = 0$ for $n \geq 5$, and $\overline{c}_{2}$ is defined in Lemma 6.4.

Proof

We will follow the proof of Propositions 3.1, 3.3, and 3.4, and we will precise the estimate of some integrals. In fact, in this case, we will use the function $\psi _{i}:= \lambda _{i} \partial \delta _{i} / \partial \lambda _{i}$. Note that (34), (35), (36), (37), (38), and (39) hold in this case since they are based on the fact that $| \psi _{i} | \leq c \delta _{i}$, which is also true with the new $\psi _{i}$. Some changes are needed for (40). In fact, using Lemma 6.7, we have

$$\begin{aligned} \frac{1}{p} \int _{\Omega }\nabla \psi _{i} \nabla v - \int _{ \partial \Omega } \frac{\partial \psi _{i} }{\partial \nu } v = 0 + O \biggl( \frac{1}{\lambda _{i} ^{(n-2)/2} } \int _{\partial \Omega } \frac{ \vert v \vert }{ \vert x - a _{i} \vert ^{n-1} } \biggr) = O \biggl( \frac{1}{ ( \lambda _{i} d_{i} ) ^{(n-2)/2} } \Vert v \Vert \biggr). \end{aligned}$$

Furthermore, (41) and (42) also hold true, but (43) becomes as follows (see (3.4) of [6]):

$$\begin{aligned} \int _{\Omega } \delta _{i} ^{ p + \varepsilon } \lambda _{i} \frac{ \partial \delta _{i}}{ \partial \lambda _{i}} = \lambda _{i} ^{ \varepsilon (n-2)/2} c _{1} \varepsilon + O \biggl( \varepsilon ^{2} + \frac{ 1}{ ( \lambda _{i} d_{i}) ^{n}} \biggr), \end{aligned}$$

where

$$\begin{aligned} c_{1}:= \frac{(n-2)^{2} }{4} c_{0}^{2n/(n-2)} \int _{\mathbb{R}^{n}} \frac{ \vert x \vert ^{2} - 1 }{ (1 + \vert x \vert ^{2} )^{n+1} } \ln \bigl(1+ \vert x \vert ^{2} \bigr) \,dx > 0. \end{aligned}$$

(49)

Concerning (44), using Lemmas 6.2 and 6.4, it holds

$$\begin{aligned} \int _{\Omega }\delta _{i} ^{p+\varepsilon -1} \psi _{i} \delta _{j} & = c_{0}^{\varepsilon } \lambda _{i}^{\frac{\varepsilon (n-2)}{2}} \int _{ \mathbb{R}^{n}} \delta _{i}^{p-1} \psi _{i} \delta _{j} + O \biggl( \int _{\mathbb{R}^{n}\setminus \Omega } \delta _{i}^{p} \delta _{j} + \varepsilon \int _{\Omega }\delta _{j} \delta _{i}^{\frac{n+2}{n-2}} \ln \bigl(1+\lambda _{i}^{2} \vert x-a_{i} \vert ^{2}\bigr) \biggr) \\ &=c_{0}^{\varepsilon }\lambda _{i}^{\frac{\varepsilon (n-2)}{2}} \frac{ {\overline{c}_{2}} }{p} \lambda _{i} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}}+O \biggl( \varepsilon _{ij}^{\frac{n}{n-2}} \ln \varepsilon _{ij}^{-1} + \frac{ 1 }{ (\lambda _{i} d_{i} )^{{(n+2)} / {2}}} \frac{ 1 }{ ( \lambda _{j} d_{j} )^{{(n - 2)} / {2}}} + \varepsilon \varepsilon _{ij} \biggr). \end{aligned}$$

In addition, (45), (46), and (47) hold. However, (48) becomes

$$\begin{aligned} \int _{\Omega }\delta _{j}^{p}\psi _{i} = \int _{\mathbb{R}^{n}} \delta _{j}^{p}\psi _{i} + O \biggl( \int _{ \mathbb{R}^{n} \setminus \Omega } {\delta _{j}^{p}\delta _{i}} \biggr) = {\overline{c}_{2}} { \lambda _{i}}\frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}} + O \biggl( \varepsilon _{ij}^{\frac{n}{n-2}} \ln \varepsilon _{ij}^{-1} + \frac{1 }{ ( \lambda _{j} d_{j} ) ^{\frac{n+2}{2}} ( \lambda _{i} d_{i} ) ^{\frac{ n-2 }{2}}} \biggr). \end{aligned}$$

Hence the analogue of Proposition 3.1 becomes

$$\begin{aligned} \int _{\Omega } \vert u \vert ^{p-1+\varepsilon }u {\lambda _{i}} \frac{\partial \delta _{i}}{\partial \lambda _{i}} = {}& \overline{c}_{2} \sum _{j\neq i} \alpha _{j} {\lambda _{i}} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}} \bigl( \alpha _{i}^{p-1+\varepsilon }\lambda _{i}^{\varepsilon \frac{ n-2}{2}} + \alpha _{j}^{p-1+\varepsilon }\lambda _{j}^{ \varepsilon \frac{ n-2}{2}} \bigr) \\ &{} + O \biggl( \sum_{k\neq r}\varepsilon _{kr}^{\frac{n}{n-2}}\ln \varepsilon _{kr}^{-1} + \Vert v \Vert ^{2} + \varepsilon ^{2} + \frac{1}{ ( \lambda _{i} d_{i} ) ^{ n - 2 } } + \sum \frac{1}{ ( \lambda _{k} d_{k} ) ^{n} } \biggr). \end{aligned}$$

(50)

For the analogue of Proposition 3.4, it holds (since $v \in E_{a,\lambda }$)

$$\begin{aligned} \int _{\Omega }\nabla u \nabla \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} & = \sum_{j=1}^{N} \alpha _{j} \biggl( \int _{\Omega }\delta _{j} ^{p} \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} + \int _{ \partial \Omega } \frac{\partial \delta _{j}}{\partial \nu } \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} \biggr) \\ &=\alpha _{i} \int _{\mathbb{R}^{n}\setminus \Omega } \delta _{i} ^{p} \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} + \sum_{j\neq i} \alpha _{j} \int _{\Omega }\delta _{j} ^{p} \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} + \sum_{j=1}^{N} \alpha _{j} \int _{\partial \Omega } \frac{\partial \delta _{j}}{\partial \nu } \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}}. \end{aligned}$$

Notice that, using Lemma 6.7, we have

$$\begin{aligned} \int _{\partial \Omega } \biggl\vert \frac{\partial \delta _{j}}{\partial \nu } \biggr\vert \lambda _{i} \biggl\vert \frac{\partial \delta _{i} }{\partial \lambda _{i}} \biggr\vert & \leq \frac{ c }{ ( \lambda _{i} \lambda _{j} ) ^{ (n - 2)/2 } } \int _{ \partial \Omega } \frac{ 1 }{ \vert x - a_{j} \vert ^{ n-1 } } \frac{ 1 }{ \vert x - a_{i} \vert ^{ n-2 } } \\ & \leq \frac{ c }{ ( \lambda _{i} \lambda _{j} ) ^{ ( n - 2 ) / { 2 } }} \biggl( \int _{\partial \Omega } \frac{ 1 }{ \vert x - a_{j} \vert ^{ 2( n-1)^{2} / { n } }} \biggr)^{ \frac{ n}{2n-2 }} \biggl( \int _{\partial \Omega } \frac{ 1 }{ \vert x - a_{i} \vert ^{ 2 n - 2 } } \biggr) ^{\frac{ n-2 }{ 2n-2 }} \\ & \leq \frac{ c }{ (\lambda _{j} d_{j} ) ^{(n-2) / 2} } \frac{ c }{ (\lambda _{i} d_{i} ) ^{(n-2) / 2} }. \end{aligned}$$

Thus, using Lemma 6.4, we obtain

$$\begin{aligned} \int _{\Omega }\nabla u \nabla \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} ={}& \sum_{ j \neq } \alpha _{j} \overline{c}_{2} \lambda _{i} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i} } \\ &{} + O \biggl( \sum_{j\neq i} \varepsilon _{ij}^{\frac{n}{n-2}} \ln \varepsilon _{ij}^{-1} +\frac{ 1 }{ ( \lambda _{i} d_{i} )^{(n-2)/2} } \sum \frac{1}{ (\lambda _{k} d_{k} ) ^{(n-2) / 2} } \biggr). \end{aligned}$$

(51)

It remains the analogue of Proposition 3.3. Using Lemma 6.2, we have

$$\begin{aligned} \int _{\Omega }u \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} = \alpha _{i} \int _{\Omega }\delta _{i} \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} + O \biggl( \int _{ \Omega } \vert v \vert \delta _{i} + \sum _{ j \neq i } \int _{\Omega }\delta _{j} \delta _{i} \biggr). \end{aligned}$$

The last integral is computed in Lemma 6.6, and the second one is computed in (31). Concerning the first one, it depends on the dimension n. If $n \geq 5$, then it holds

$$\begin{aligned} \int _{\Omega }\delta _{i} \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} ={}& \frac{n-2}{2} c_{0} ^{2} \int _{\mathbb{R}^{n}} \lambda _{i} ^{n-2} \frac{ 1 - \lambda _{i} ^{2} \vert x - a_{i} \vert ^{2} }{ ( 1 + \lambda _{i} ^{2} \vert x - a_{i} \vert ^{2} )^{n-1} } + O \biggl( \int _{\mathbb{R}^{n} \setminus \Omega } \frac{ 1 }{ \lambda _{i} ^{n-2} } \frac{ 1 }{ \vert x-a_{i} \vert ^{2n-4}} \biggr) \\ = {}&{-} \frac{ c(n) }{ \lambda _{i} ^{2} } + O \biggl( \frac{ 1 }{ \lambda _{i} ^{2} ( \lambda _{i} d_{i} ) ^{n-4} } \biggr) \\ &\quad \text{with } c(n):= \frac{n-2}{2} c_{0} ^{2} \int _{ \mathbb{R}^{n}} \frac{ \vert x \vert ^{2} - 1 }{ ( 1 + \vert x \vert ^{2} )^{n-1} } > 0 . \end{aligned}$$

(52)

However, for $n = 4$, let $r, R >0$ be such that $B(a_{i}, r ) \subset \Omega \subset B(a_{i},R)$. (Note that, if $a_{i}$ is in a compact set of Ω, then r will be independent of $a_{i}$). It holds

$$\begin{aligned} \int _{\Omega }\delta _{i} \lambda _{i} \frac{\partial \delta _{i} }{\partial \lambda _{i}} & = \frac{n-2}{2} c_{0} ^{2} \int _{ B(a_{i},r) } \lambda _{i} ^{2} \frac{ 1 - \lambda _{i} ^{2} \vert x - a_{i} \vert ^{2} }{ ( 1 + \lambda _{i} ^{2} \vert x - a_{i} \vert ^{2} )^{ 3 } } + O \biggl( \int _{ B(a_{i}, R) \setminus B(a_{i},r) } \frac{ 1 }{ \lambda _{i} ^{ 2 } } \frac{ 1 }{ \vert x - a_{i} \vert ^{4}} \biggr) \\ & = - c(n) \frac{ \ln \lambda _{i} }{ \lambda _{i} ^{2} } + O \biggl( \frac{ 1 }{ \lambda _{i} ^{ 2 } } \biggr) \quad\text{with } c(n):= \frac{n-2}{2}\mathrm{meas} \bigl(S^{3}\bigr) c_{0} ^{2}. \end{aligned}$$

(53)

Finally, combining (50), (51), (52), and (53), the proof of Proposition 3.7 follows. □

Lastly, we give the following expression involving the gluing parameters $\alpha _{i}'s$. Namely, we have

Proposition 3.8

Let $n \geq 4$ and $u = \sum_{ j \leq N} \alpha _{j} \delta _{a_{j},\lambda _{j}} + v$ be such that $(\alpha,a,\lambda ) \in \mathcal{V}(N,\varepsilon,\eta )$ and $v \in E_{a,\lambda }$. Then, for $1\leq i\leq N$, the following fact holds:

$$\begin{aligned} &\int _{\Omega }\nabla u \cdot \nabla \delta _{i} + \mu \int _{\Omega }u \delta _{i} - \int _{\Omega } \vert u \vert ^{p-1+\varepsilon } u \delta _{i} \\ &\quad= \alpha _{i} S_{n} \bigl( 1 - \alpha _{i} ^{ p-1 + \varepsilon } \lambda _{i} ^{\varepsilon \frac{n-2}{2} } \bigr) \\ & \qquad{}+ O \biggl( \varepsilon + \Vert v \Vert ^{2} + \sum \varepsilon _{kr} + \frac{\ln ^{\sigma _{n}}\lambda _{i} }{\lambda _{i} ^{2}} + \sum \frac{ 1 }{ (\lambda _{k} d_{k}) ^{ n - 2 }} \biggr). \end{aligned}$$

Proof

The proof can be done as the previous ones, and it is more easy. Hence we omit it. □

4 Proof of the main results

This section is devoted to the proof of the main results of the paper. Their proof is basically based on the precise estimates made in Sect. 3. We start by excluding the existence of solutions that concentrate at a single interior point.

4.1 Proof of Theorem 1.1

We argue by contradiction, assume that such a sequence of solutions $(u_{\varepsilon })$ exists. Thus the solution $u_{\varepsilon }$ will have the form (3) that is $u_{\varepsilon }= \alpha _{\varepsilon }\delta _{a_{\varepsilon }, \lambda _{\varepsilon }} + v _{\varepsilon }$ and properties (4) are satisfied. Furthermore, Lemma 2.1 and Proposition 2.4 hold true with $N=1$. Hence, $(\alpha _{\varepsilon }, a_{\varepsilon }, \lambda _{\varepsilon }) \in \mathcal{V}(1,\varepsilon,\eta )$ for small $\eta > 0$ and $v_{\varepsilon }\in E_{a_{\varepsilon },\lambda _{\varepsilon }}$. Thus, using Propositions 3.7 and 2.4, we obtain the following:

$$\begin{aligned} 0 = - c_{1} \varepsilon - c(n) \mu \frac{ \ln ^{\sigma _{n}} (\lambda _{\varepsilon }) }{ \lambda _{\varepsilon }^{2}} + o \biggl( \varepsilon + \frac{ \ln ^{\sigma _{n}} (\lambda _{\varepsilon }}{\lambda _{\varepsilon }^{2}} \biggr), \end{aligned}$$

which gives a contradiction. Hence the proof is completed.

In the next subsection, we give a partial characterization of the solutions that concentrate at interior points if there exist.

4.2 Proof of Theorem 1.2

Let $(u_{\varepsilon })$ be a sequence of solutions of $(\mathcal{P}_{\varepsilon })$ satisfying the assumptions of the theorem. Thus the solution $u_{\varepsilon }$ will have the form (3), that is, $u_{\varepsilon }= \sum_{k=1}^{N} \alpha _{k,\varepsilon } \delta _{a_{k, \varepsilon }, \lambda _{k,\varepsilon }} + v _{\varepsilon }$ and properties (4) are satisfied. Furthermore, Lemma 2.1 and Proposition 2.4 hold true. In addition, Propositions 3.5–3.7 hold and the left-hand side in each proposition is equal to 0. For the sake of simplicity, we will omit the index ε of the variables. Furthermore, without loss of the generality, we can order the $\lambda _{i}$s as follows:

$$\begin{aligned} \lambda _{1} \leq \lambda _{2} \leq \cdots \leq \lambda _{N}. \end{aligned}$$

First, multiplying Proposition 3.7 with $2^{i}$ and summing over $i =1, \ldots, N$, it holds (by using Proposition 2.4 and Assertion (3) of Lemma 6.5)

$$\begin{aligned} \sum_{k\neq r} \varepsilon _{kr} \leq c \biggl( \varepsilon + \frac{ \ln ^{\sigma _{n}} (\lambda _{1}) }{\lambda _{1}^{2}} \biggr),\quad \text{where } \sigma _{n} = \textstyle\begin{cases} 1 & \text{if } n = 4, \\ 0 & \text{if } n \geq 5. \end{cases}\displaystyle \end{aligned}$$

(54)

First, we prove Assertion (i) in the theorem, arguing by contradiction. Let j be such that $\lambda _{j} / \lambda _{1}$ is bounded and assume that $| a_{j} - a_{k} | \geq c > 0$ for each $k\neq j$. Thus, we derive that $\varepsilon _{kj} \leq c/( \lambda _{j} \lambda _{k})^{(n-2)/2} \leq c / \lambda _{1}^{n-2}$ for each $k \neq j$. Now, writing Proposition 3.7 with $i=j$ and recalling that the left-hand side is 0, we obtain

$$\begin{aligned} - c_{1} \varepsilon - c(n) \mu \frac{ \ln ^{\sigma _{n}} ( \lambda _{j} ) }{\lambda _{j} ^{2}} = o \biggl( \varepsilon + \frac{ \ln ^{\sigma _{n}} (\lambda _{1}) }{\lambda _{1}^{2}} \biggr), \end{aligned}$$

which gives a contradiction since $\lambda _{j}$ and $\lambda _{1}$ are of the same order. Thus Assertion (i) follows.

Second, we focus on the proof of Assertion (ii). In the sequel, we therefore assume that $n\geq 5$. Now, using Proposition 2.4, we get

$$\begin{aligned} \Vert v_{\varepsilon } \Vert ^{2} \leq{}& C \Biggl( \varepsilon ^{2} + \sum_{i=1}^{N} \frac{1}{\lambda _{i} ^{n-2}} + \sum_{1\leq i,j\leq N, j \neq i } \bigl(\varepsilon _{ij} ^{\frac{n+2}{n-2}} \bigl(\ln \varepsilon _{ij}^{-1} \bigr)^{\frac{n+2}{n}}+ \varepsilon _{ij}^{2} \bigr) \Biggr) \\ &{} + C\sum_{ i = 1 }^{N} \textstyle\begin{cases} \lambda _{i} ^{-4 } \ln ^{4/3} \lambda _{i} & \text{if } n = 6, \\ \lambda _{i}^{ -\min (4,n-2)} & \text{if } n \neq 6. \end{cases}\displaystyle \end{aligned}$$

(55)

Note that for $n\geq 6$ we have $2n/(n-2) < n-2$ and for $n\geq 5$ we have $2>n/(n-2)$. This implies that

$$\begin{aligned} \varepsilon ^{2}=o\bigl(\varepsilon ^{n/(n-2)} \bigr) \quad\forall n\geq 5 \quad\text{and}\quad \lambda _{i}^{2-n}= o \bigl( \lambda _{1} ^{-2n/(n-2)}\bigr)\quad \forall n\geq 6. \end{aligned}$$

(56)

Now, using (54), (55), and (56), the estimate of $\| v_{\varepsilon }\|^{2}$ can be written as

$$\begin{aligned} \Vert v _{\varepsilon } \Vert ^{2} = O(R_{v})\quad \text{where } R_{v}:= o\bigl( \varepsilon ^{n/(n-2)} \bigr) + \textstyle\begin{cases} O(1 / \lambda _{1} ^{3} ) & \text{if } n = 5, \\ o( 1 / \lambda _{1} ^{2n/(n-2)} ) & \text{if } n \geq 6. \end{cases}\displaystyle \end{aligned}$$

Furthermore, the remaining term in Proposition 3.5 can be written as

$$\begin{aligned} R_{35,i} = o \biggl( \varepsilon ^{(n-1)/(n-2)} + \frac{1}{ \lambda _{1} ^{2(n-1)/(n-2)}} + \sum_{j\neq i } \frac{1}{\lambda _{i}} \biggl\vert \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \biggr\vert \biggr)+O \biggl( \frac{1}{\lambda _{i}}\sum_{j\neq i} \varepsilon _{ij} \biggr). \end{aligned}$$

(57)

Now, we will focus on Assertion (ii). Arguing by contradiction, assume that $\lambda _{N} / \lambda _{1}$ is bounded. From the smallness of the $\varepsilon _{ij}'s$, we deduce that $\lambda _{i}|a_{i}-a_{j}|\to \infty$ for each $j\neq i$. Let $\gamma:= \min \{ | a_{j} - a_{1} |: j \neq i\} >0$. Since the $\lambda _{k}'s$ are of the same order, without loss of generality, we can assume that $\gamma =|a_{1}-a_{i_{0}}|$ for some $i_{0}\neq 1$. Let $N_{1}:= \{ j: |a_{j} - a_{1} | \to 0\}$. Note that $1 \in N_{1}$ and Assertion (i) implies that $N_{1}$ contains at least another index. Since we have assumed that $\lambda _{N} / \lambda _{1}$ is bounded, it follows that

$$\begin{aligned} \varepsilon _{1j} \leq \frac{1}{(\lambda _{1} \lambda _{j})^{(n-2)/2} \vert a_{1} - a_{j} \vert ^{n-2} } \leq \frac{c}{(\lambda _{1} \lambda _{i_{0}})^{(n-2)/2} \vert a_{1} - a_{i_{0}} \vert ^{n-2} } \leq c \varepsilon _{1i_{0}}, \quad\forall 2\leq j \leq N. \end{aligned}$$

(58)

Now, we need to introduce the points that are very close to $a_{1}$. Let us define $N'_{2}:= \{ j \in N_{1}: | a_{j} - a_{1} | / \gamma \to \infty \}$ and $N_{2}:= N_{1} \setminus N'_{2}$. We remark that the $\varepsilon _{ij}$s, for $i, j \in N_{2}$ with $i \neq j$, are of the same order (in the sense that $c \leq \varepsilon _{ij} / \varepsilon _{kr} \leq c'$ for each $i, j, k, r \in N_{2}$).

Let a̅ be such that $\sum_{j \in N_{2}} (a_{j} - \overline{a}) = 0$. It is easy to see that $| a_{j} - \overline{a} | \leq c \gamma$ for each $j \in N_{2}$. Hence it follows that (by using the fact that $\lambda _{N} / \lambda _{1}$ is bounded and Assertion (8) of Lemma 6.5)

$$\begin{aligned} \lambda _{i} \vert a_{i} - \overline{a} \vert \leq c \sqrt{ \lambda _{1} \lambda _{i_{0}} } \vert a_{i_{0}} - a_{1} \vert \leq c \varepsilon _{1i_{0}} ^{-1/(n-2)}\quad \forall i \in N_{2}. \end{aligned}$$

(59)

Combining Propositions 3.8, 2.4, and 3.5 and using (57), we derive that, for $n \geq 5$ and for each $i \in N_{2}$,

$$\begin{aligned} &\frac{1}{\lambda _{i}} \sum_{j \neq i} \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \\ &\quad= o \biggl( \varepsilon ^{(n-1)/(n-2)} + \frac{ 1 }{ \lambda _{1} ^{ 2(n-1)/(n-2)}} + \frac{1}{\lambda _{i}} \sum_{j \neq i} \biggl\vert \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \biggr\vert \biggr)+O \biggl( \frac{1}{\lambda _{i}}\sum _{j\neq i} \varepsilon _{ij} \biggr) \quad\forall i \leq N. \end{aligned}$$

(60)

Since the $\lambda _{j}$s are of the same order, using (54) and Assertions (1) and (8) of Lemma 6.5, we get

$$\begin{aligned} &\frac{1}{\lambda _{i}} \sum_{j \neq i} \biggl\vert \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \biggr\vert \leq \frac{c }{\lambda _{i} \vert a_{i} - a_{j} \vert } \varepsilon _{ij} \leq c \varepsilon _{ij}^{(n-1)/(n-2)} \leq c \varepsilon ^{(n-1)/(n-2)} + \frac{ c }{ \lambda _{1} ^{ 2(n-1)/(n-2)}}, \\ &\frac{1}{\lambda _{i}}\varepsilon _{ij}\leq \frac{\gamma \varepsilon _{ij}}{\sqrt{\lambda _{1}\lambda _{i_{0}}} \vert a_{1}-a_{i_{0}} \vert } \leq c \gamma \bigl(\varepsilon _{1i_{o}}^{1/(n-2)}\varepsilon _{ij} \bigr)=o \bigl(\varepsilon _{1i_{o}}^{(n-1)/(n-2)}+ \varepsilon _{ij}^{(n-1)/(n-2)} \bigr). \end{aligned}$$

Multiplying (60) by $\lambda _{i} ( \overline{a} - a_{i})$ and summing over $i \in N_{2}$, we obtain

$$\begin{aligned} \sum_{i \in N_{2}} \sum _{j \neq i} \frac{\partial \varepsilon _{ij}}{\partial a_{i}} ( \overline{a} - a_{i} ) = o \biggl( \sum_{i \in N_{2}} \lambda _{i} \vert a_{i} - \overline{a} \vert \biggl( \varepsilon ^{(n-1)/(n-2)} + \frac{ 1 }{ \lambda _{1} ^{ 2(n-1)/(n-2)}} \biggr) \biggr). \end{aligned}$$

(61)

To proceed further, we split the above sum on j into three blocks.

Block 1: $i, j \in N_{2}$ with $j \neq i$. In this group, using Assertions (1) and (8) of Lemma 6.5, we observe that

$$\begin{aligned} \frac{\partial \varepsilon _{ij}}{\partial a_{i}} (\overline{a} - a_{i} ) + \frac{\partial \varepsilon _{ij}}{\partial a_{j}} ( \overline{a} - a_{j} )& = \frac{\partial \varepsilon _{ij}}{\partial a_{i}} ( a_{j} - a_{i} ) \\ &= (n-2) \lambda _{i} \lambda _{j} \vert a_{i} - a_{j} \vert ^{2} \varepsilon _{ ij }^{\frac{n}{n-2}} = (n-2) \varepsilon _{ij} \bigl(1+o(1)\bigr). \end{aligned}$$

(62)

Block 2: $i \in N_{2}$ and $j \notin N_{1}$, that is, $|a_{i}-a_{j}|\geq c>0$. In this case, using Assertion (1) of Lemma 6.5, we obtain

$$\begin{aligned} \biggl\vert \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \biggr\vert \vert \overline{a} - a_{i} \vert = (n-2) \lambda _{i} \lambda _{j} \vert a_{i} - a_{j} \vert \vert \overline{a} - a_{i} \vert \varepsilon _{ij} ^{n/(n-2)} \leq \frac{ \vert \overline{a} - a_{i} \vert }{( \lambda _{i} \lambda _{j} )^{(n-2)/2}} = o( \varepsilon _{1i_{0}}). \end{aligned}$$

(63)

Block 3: $i \in N_{2}$ and $j \in N_{1} \setminus N_{2}$. In this group, using Assertion (8) of Lemma 6.5, the fact that $|a_{i} -\bar{a}|\leq C \gamma$ for each $i\in N_{2}$ and $|a_{i}-a_{j}|\gg |a_{1}-a_{i_{0}}|$, we get

$$\begin{aligned} \biggl\vert \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \biggr\vert \vert \overline{a} - a_{i} \vert \leq \frac{ c \vert \overline{a} - a_{i} \vert }{( \lambda _{i} \lambda _{j} )^{(n-2)/2} \vert a_{i} - a_{j} \vert ^{n-1} } = o \biggl( \frac{ \vert \overline{a} - a_{i} \vert }{( \lambda _{1} \lambda _{i_{0}} )^{(n-2)/2} \vert a_{1} - a_{i_{0}} \vert ^{n-1} } \biggr) = o( \varepsilon _{1i_{0}} ). \end{aligned}$$

(64)

Combining estimates (62), (63), (64), and (61), we deduce that

$$\begin{aligned} \sum_{ k\neq j, k, j \in N_{2}} \varepsilon _{kj} \leq o \biggl( \varepsilon _{1i_{0}}^{-1/(n-2)} \biggl( \varepsilon ^{(n-1)/(n-2)} + \frac{ 1 }{ \lambda _{1} ^{ 2(n-1)/(n-2)}} \biggr) \biggr), \end{aligned}$$

which implies

$$\begin{aligned} \varepsilon _{1i_{0}} = o \biggl( \varepsilon + \frac{ 1 }{ \lambda _{1} ^{ 2} } \biggr). \end{aligned}$$

(65)

Putting (58) and (65) in Proposition 3.7 with $i=1$ and using the fact that $u_{\varepsilon }$ is a solution of $(\mathcal{P}_{\varepsilon })$ (which implies that the left-hand side of the proposition is 0), we obtain

$$\begin{aligned} - c_{1} \varepsilon - c(n) \frac{\mu}{\lambda _{1}^{2}} = o \biggl( \varepsilon + \frac{ 1 }{ \lambda _{1} ^{ 2} } \biggr), \end{aligned}$$

(66)

which presents a contradiction.

Hence the proof of the theorem is completed.

In the next subsection, we use Theorem 1.2 and the precise estimates of Sect. 3 to exclude the case of the existence of solutions with two or three interior blow-up points.

4.3 Proof of Theorem 1.3

We argue by contradiction. Assume that such a sequence of solutions $(u_{\varepsilon })$ exists. Thus the solution $u_{\varepsilon }$ will have the form (3) that is $u_{\varepsilon }= \sum_{k=1}^{N}\alpha _{k,\varepsilon } \delta _{a_{k, \varepsilon }, \lambda _{k,\varepsilon }} + v _{\varepsilon }$ with $N\in \{2,3\}$ and properties (4) are satisfied. Furthermore, Lemma 2.1 and Proposition 2.4 hold true. As in the previous proof, without loss of the generality, we can assume that $\lambda _{1} \leq \cdots \leq \lambda _{N}$.

We first prove the theorem in the case of two interior blow-up points.

Proof of Theorem 1.3 in the case of $N=2$ and $n \geq 5$

The proof will be decomposed into three steps. The first one is a direct consequence of Theorem 1.2.

Step 1. $\lambda _{2} / \lambda _{1} \to \infty$ and $| a_{1} - a_{2} | \to 0$.

The second one is as follows.

Step 2. There exists a positive constant $\eta _{1} > 0$ such that $\lambda _{1} | a_{1} - a_{2} | \geq \eta _{1}$.

To prove Step 2, arguing by contradiction, we assume that $\lambda _{1} | a_{1} - a_{2} | \to 0$ as $\varepsilon \to 0$. Thus, using Assertion (7) of Lemma 6.5, Proposition 3.7 with $i=1$ implies

$$\begin{aligned} - c_{1} \varepsilon - c(n) \frac{ \mu }{ \lambda _{1} ^{2}} - \overline{c}_{2} \frac{n-2}{2} \varepsilon _{12} = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} + \varepsilon _{12} \biggr), \end{aligned}$$

which cannot occur (since $\varepsilon >0$ and $\mu >0$). Hence Step 2 follows.

Step 3. Proof of the theorem in the case mentioned above: on the one hand, using Assertions (1) and (4) of Lemma 6.5, we get

$$\begin{aligned} \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{1}} \biggr\vert \geq \frac{ c }{ \lambda _{1} \vert a_{1} - a_{2} \vert } \varepsilon _{12} \geq c \sqrt{ \frac{\lambda _{2}}{\lambda _{1}}} \varepsilon _{12} ^{ (n-1)/(n-2)}. \end{aligned}$$

(67)

On the other hand, applying Proposition 3.5 and using (57), we obtain

$$\begin{aligned} \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{1}} \biggr\vert \leq c R_{35,1} = o \biggl( \varepsilon ^{(n-1)/(n-2)} + \frac{1}{ \lambda _{1} ^{2(n-1)/(n-2)}} + \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{1}} \biggr\vert \biggr) + O \biggl( \frac{1}{\lambda _{1}} \varepsilon _{12} \biggr). \end{aligned}$$

However, using (67), we have

$$\begin{aligned} \frac{1}{\lambda _{1}} \varepsilon _{12}= \vert a_{1}-a_{2} \vert \frac{1}{\lambda _{1} \vert a_{1}-a_{2} \vert }\varepsilon _{12}=o \biggl( \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{1}} \biggr\vert \biggr). \end{aligned}$$

We derive that

$$\begin{aligned} \varepsilon _{12} = o \biggl( \varepsilon + \frac{1}{\lambda _{1}^{2}} \biggr). \end{aligned}$$

Putting this information in Proposition 3.7 with $i=1$, we derive (66), which gives a contradiction. Hence the proof of the theorem is complete in the case of $N=2$ and $n \geq 5$. □

Proof of Theorem 1.3 in the case of $N=3$ and $n \geq 6$

To make the proof clearer, we will split it into several claims. The first one is a direct consequence of Theorem 1.2.

Claim 1. $\lambda _{3} / \lambda _{1} \to \infty$, there exists $k \in \{2,3\}$ such that $| a_{1} - a_{k} | \to 0$ and (54) holds true.

Before stating the second claim, we notice that, since $u_{\varepsilon }$ is a solution of $(\mathcal{P}_{\varepsilon })$, the left-hand side of Propositions 3.5, 3.7, and 3.8 becomes 0. Thus, using (54), Propositions 2.4, 3.5, and 3.7, we obtain

$$\begin{aligned} & (E_{i})\quad - c_{1} \varepsilon - c(n) \frac{ \mu }{ \lambda _{i} ^{2}} - \overline{c}_{2} \sum_{j \neq i} \lambda _{i} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}} = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr) \quad\forall 1 \leq i \leq N, \\ & (F_{i})\quad - \overline{c}_{2} \sum _{j \neq i} \frac{1}{\lambda _{i}} \frac{\partial \varepsilon _{ij}}{\partial a_{i}}\\ &\phantom{(F_{i})\quad}\quad = O \biggl( \varepsilon ^{2} + \frac{1}{\lambda _{1} ^{4}} \biggr) + o \biggl( \sum _{j \neq i} \frac{1}{\lambda _{i}} \biggl\vert \frac{\partial \varepsilon _{ij}}{\partial a_{i}} \biggr\vert \biggr)+ O \biggl( \frac{1}{\lambda _{i}} \sum _{j\neq i}\varepsilon _{ij} \biggr)\quad \forall 1 \leq i \leq N. \end{aligned}$$

The second claim is as follows.

Claim 2. There exists a positive constant $\eta _{1}$ such that $\lambda _{1} | a_{1} - a_{2} | \geq \eta _{1}$.

To prove Claim 2, arguing by contradiction, we assume that $\lambda _{1} | a_{1} - a_{2} | \to 0$. The smallness of $\varepsilon _{12}$ implies that $\lambda _{1}/ \lambda _{2} \to 0$. Computing $(E_{3})-(E_{1})$ and using Lemma 6.5, it holds

$$\begin{aligned} c(n)\frac{\mu}{\lambda _{1}^{2}} + \frac{n-2}{2} \overline{c}_{2} \varepsilon _{12} + \frac{n-2}{2}\overline{c}_{2} \varepsilon _{23} + \overline{c}_{2} \biggl( - \lambda _{3} \frac{\partial \varepsilon _{13}}{\partial \lambda _{3}} + \lambda _{1} \frac{\partial \varepsilon _{13}}{\partial \lambda _{1}} \biggr) = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr). \end{aligned}$$

Now, using Assertion (2) of Lemma 6.5, we derive that

$$\begin{aligned} \frac{1}{\lambda _{1}^{2}}; \varepsilon _{12}; \varepsilon _{23} = o( \varepsilon ). \end{aligned}$$

Putting this information in $(E_{2})$, we obtain a contradiction. Hence the proof of this claim is complete.

Next, we prove the following claim.

Claim 3. There exists a positive constant $\eta _{2}$ such that $\lambda _{1} | a_{1} - a_{3} | \geq \eta _{2}$.

Arguing by contradiction, assume that $\lambda _{1} | a_{1} - a_{3} | \to 0$. Using Claim 2, we see that $| a_{1} - a_{3} | / | a_{1} - a_{2} | \to 0$. We distinguish two cases.

First case: $\lambda _{2} / \lambda _{1} \to \infty$. Observe that, in this case, Claim 2 implies that $\lambda _{2} | a_{1} - a_{2} | \to \infty$. Therefore $\lambda _{2} | a_{2} - a_{3} | \to \infty$ (since $| a_{1} - a_{3} | = o( | a_{1} - a_{2} |)$). Using Lemma 6.5, $(E_{2})-(E_{1})$ implies

$$\begin{aligned} c(n)\frac{\mu}{\lambda _{1}^{2}} + \frac{n-2}{2} \overline{c}_{2} \varepsilon _{13} + \frac{n-2}{2}\overline{c}_{2} \varepsilon _{23} + \overline{c}_{2} \biggl( - \lambda _{2} \frac{\partial \varepsilon _{12}}{\partial \lambda _{2}} + \lambda _{1} \frac{\partial \varepsilon _{12}}{\partial \lambda _{1}} \biggr) = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr). \end{aligned}$$

Using Assertion (2) of Lemma 6.5, we derive that

$$\begin{aligned} \frac{1}{\lambda _{1}^{2}}; \varepsilon _{13}; \varepsilon _{23} = o( \varepsilon ). \end{aligned}$$

Putting this information in $(E_{3})$, we obtain a contradiction. Hence the proof of Claim 3 follows in this case.

Second case: $\lambda _{2} / \lambda _{1}$ is bounded. In this case, the smallness of $\varepsilon _{12}$ implies that $\lambda _{j} | a_{1} - a_{2} | \to \infty$ for $j = 1,2$. Therefore $\lambda _{2} | a_{2} - a_{3} | \to \infty$ (since $| a_{1} - a_{3} | = o( | a_{1} - a_{2} |)$). Note that, using Lemma 6.5, we obtain

$$\begin{aligned} & \varepsilon _{23} \leq \frac{1}{ (\lambda _{2} \lambda _{3} \vert a_{2} - a_{3} \vert ^{2})^{(n-2)/2}} \leq c \biggl( \frac{\lambda _{1}}{\lambda _{3}} \biggr)^{(n-2)/2} \frac{1}{ (\lambda _{2} \lambda _{1} \vert a_{2} - a_{1} \vert ^{2})^{(n-2)/2}} \leq c \varepsilon _{13} \varepsilon _{12}, \\ & \frac{1}{\lambda _{2}} \biggl\vert \frac{\partial \varepsilon _{23}}{\partial a_{2}} \biggr\vert = (n-2) \lambda _{3} \vert a_{2} - a_{3} \vert \varepsilon _{23}^{n/(n-2)} \leq \frac{ c \varepsilon _{23} }{ \lambda _{2} \vert a_{2} - a_{3} \vert } \leq \frac{ c \varepsilon _{23} }{ \lambda _{2} \vert a_{2} - a_{1} \vert } \leq C\varepsilon _{12}^{1/(n-2)} \varepsilon _{23}. \end{aligned}$$

Putting this information in $(F_{2})$ and using (54), we obtain

$$\begin{aligned} \frac{1}{\lambda _{2}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{2}} \biggr\vert \leq \frac{c}{\lambda _{2}} \biggl\vert \frac{\partial \varepsilon _{23}}{\partial a_{2}} \biggr\vert + c \biggl( \varepsilon ^{2} + \frac{1}{\lambda _{1} ^{4}} \biggr) \leq c \biggl( \varepsilon ^{2} + \frac{1}{\lambda _{1} ^{4}} \biggr). \end{aligned}$$

On the other hand, using Lemma 6.5 and the fact that $\lambda _{2}/\lambda _{1}$ is bounded, we get

$$\begin{aligned} \frac{1}{\lambda _{2}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{2}} \biggr\vert = (n-2) \lambda _{1} \vert a_{2} - a_{1} \vert \varepsilon _{12}^{n/(n-2)} \geq c \varepsilon _{12}^{(n-1)/(n-2)}. \end{aligned}$$

Thus

$$\begin{aligned} \varepsilon _{12} = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr). \end{aligned}$$

(68)

Putting this information in $(E_{1})$, using the fact that $\lambda _{1} |a_{1}-a_{3}|\to 0$ and Lemma 6.5, we obtain

$$\begin{aligned} - c_{1} \varepsilon - c(n) \frac{ \mu }{ \lambda _{1} ^{2}} - \frac{n-2}{2} \bar{c}_{2}\varepsilon _{13} = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr), \end{aligned}$$

which gives a contradiction. Hence the proof of Claim 3 also follows in this case. This completes the proof of Claim 3.

Now, we state and prove Claim 4.

Claim 4. There exists a positive constant $\eta _{3}$ such that $| a_{1} - a_{2} | / | a_{1} - a_{3} | \geq \eta _{3}$.

Arguing by contradiction, assume that $| a_{1} - a_{2} | = o( | a_{1} - a_{3} | )$. Using Lemma 6.5, we observe that

$$\begin{aligned} &\frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{1}} \biggr\vert \geq \frac{ c \varepsilon _{ 12} }{\lambda _{1} | a_{1} - a_{2} ] } \geq c \varepsilon _{ 12} ^{(n-1)/(n-2)}, \\ &\varepsilon _{13} \leq \frac{1}{ (\lambda _{1} \lambda _{3} \vert a_{1} - a_{3} \vert ^{2})^{(n-2)/2}} = o \biggl( \frac{1}{ (\lambda _{2} \lambda _{1} \vert a_{2} - a_{1} \vert ^{2})^{(n-2)/2}} \biggr) = o( \varepsilon _{12}), \\ &\frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{13}}{\partial a_{1}} \biggr\vert \leq \frac{ c \varepsilon _{ 13} }{\lambda _{1} | a_{1} - a_{3} ] } = o \biggl( \frac{ c \varepsilon _{ 12} }{\lambda _{1} | a_{1} - a_{2} ] } \biggr). \end{aligned}$$

(69)

Thus, $(F_{1})$ implies that

$$\begin{aligned} \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{1}} \biggr\vert \bigl(1+o(1) \bigr) = O \biggl(\varepsilon ^{2} + \frac{1}{\lambda _{1} ^{4}} \biggr), \end{aligned}$$

which shows that (68) holds. Putting (69) and (68) in $(E_{1})$, we obtain (66), which gives a contradiction. Hence the proof of Claim 4 follows.

Next, we prove the following.

Claim 5. There exists a positive constant $\eta _{4} > 0$ such that $| a_{1} - a_{3} | / | a_{1} - a_{2} | \geq \eta _{4}$.

Arguing by contradiction, assume that $| a_{1} - a_{3} | = o( | a_{1} - a_{2} | )$. First, observe that

$$\begin{aligned} \varepsilon _{23} \leq \frac{1}{ (\lambda _{2} \lambda _{3} \vert a_{2} - a_{3} \vert ^{2})^{(n-2)/2}} = o \biggl( \frac{1}{ (\lambda _{2} \lambda _{1} \vert a_{2} - a_{1} \vert ^{2})^{(n-2)/2}} \biggr) = o( \varepsilon _{12}), \end{aligned}$$

(70)

where we have used in the last inequality Claim ! and Assertion (8) of Lemma 6.5.

Second, we distinguish three cases and we will prove that all these cases cannot occur.

First case: $\lambda _{2}/ \lambda _{1}$ remains bounded. Using Lemma 6.5 and (70), it holds

$$\begin{aligned} & \frac{1}{\lambda _{2}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{2}} \biggr\vert \geq \frac{ c \varepsilon _{ 12} }{\lambda _{2} | a_{1} - a_{2} ] } \geq c \varepsilon _{ 12} ^{(n-1)/(n-2)}, \\ & \frac{1}{\lambda _{2}} \biggl\vert \frac{\partial \varepsilon _{23}}{\partial a_{2}} \biggr\vert \leq \frac{ c \varepsilon _{ 23} }{\lambda _{2} \vert a_{2} - a_{3} \vert } = o \biggl( \frac{ \varepsilon _{ 12} }{\lambda _{2} \vert a_{1} - a_{2} \vert } \biggr). \end{aligned}$$

Therefore, using $(F_{2})$, we get

$$\begin{aligned} \frac{1}{\lambda _{2}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{2}} \biggr\vert \bigl(1+o(1) \bigr) = O \biggl( \varepsilon ^{2} + \frac{1}{\lambda _{1}^{4}} \biggr), \end{aligned}$$

which implies that (68) holds in this case. Now, taking $(E_{3} - E_{2} - E_{1})$ and using (70), (68), we obtain

$$\begin{aligned} c_{1} \varepsilon + c(n)\frac{ \mu }{\lambda _{1}^{2}} + c(n) \frac{ \mu }{\lambda _{2}^{2}} + \overline{c}_{2} \biggl( - \lambda _{3} \frac{\partial \varepsilon _{13}}{\partial \lambda _{3}} + \lambda _{1} \frac{\partial \varepsilon _{13}}{\partial \lambda _{1}} \biggr) = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr). \end{aligned}$$

(71)

Using Assertion (2) of Lemma 6.5, we obtain a contradiction. Hence this case cannot occur.

Second case: $\lambda _{2}/ \lambda _{1} \to \infty$ and $\varepsilon _{13} = o( \varepsilon _{12} )$. In this case, taking $(E_{2} - E_{1} - E_{3})$ and using (70), we get

$$\begin{aligned} c_{1} \varepsilon + c(n)\frac{ \mu }{\lambda _{1}^{2}} + \overline{c}_{2} \biggl( - \lambda _{2} \frac{\partial \varepsilon _{12}}{\partial \lambda _{2}} + \lambda _{1} \frac{\partial \varepsilon _{12}}{\partial \lambda _{1}} \biggr) = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr). \end{aligned}$$

Again, using Assertion (2) of Lemma 6.5, we obtain a contradiction. Hence this case cannot also occur.

Third case: $\lambda _{2}/ \lambda _{1} \to \infty$ and $\varepsilon _{13} \geq c \varepsilon _{12}$ for some positive constant c.

Using assertions (1) and (4) of Lemma 6.5, it follows that

$$\begin{aligned} \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{13}}{\partial a_{1}} \biggr\vert &\geq c \lambda _{3} \vert a_{1}-a_{3} \vert \varepsilon _{13}^{n/(n-2)} \geq \frac{c}{\lambda _{1} \vert a_{1}-a_{3} \vert } \varepsilon _{13} \\ &\geq c\sqrt{\frac{\lambda _{3}}{\lambda _{1}}} \frac{1}{\sqrt{\lambda _{1}\lambda _{3}} \vert a_{1}-a_{3} \vert } \varepsilon _{13}\gg \varepsilon _{13}^{\frac{n-1}{n-2}} \end{aligned}$$

(72)

and

$$\begin{aligned} \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{1}} \biggr\vert & \leq c \lambda _{2} \vert a_{1}-a_{2} \vert \varepsilon _{12}^{n/(n-2)} \leq \frac{c}{\lambda _{1} \vert a_{1}-a_{2} \vert }\varepsilon _{12} \leq \frac{c}{\lambda _{1} \vert a_{1}-a_{2} \vert }\varepsilon _{12} \\ &\leq \frac{c}{\lambda _{1} \vert a_{1}-a_{2} \vert }\varepsilon _{13} \ll \frac{c}{\lambda _{1} \vert a_{1}-a_{3} \vert } \varepsilon _{13} \leq \frac{c}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{13}}{\partial a_{1}} \biggr\vert , \end{aligned}$$

where we have used in the last line the fact that $|a_{1}-a_{3}|=o(|a_{1}-a_{2}|)$ and (72).

Therefore, using $(F_{1})$, we get

$$\begin{aligned} \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{13}}{\partial a_{1}} \biggr\vert \bigl(1+o(1) \bigr) = O \biggl( \varepsilon ^{2} + \frac{1}{\lambda _{1}^{4}} \biggr), \end{aligned}$$

which implies that

$$\begin{aligned} \varepsilon _{12}; \varepsilon _{13} = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr). \end{aligned}$$

(73)

Thus, putting this information in $(E_{1})$, we easily obtain a contradiction. Thus this case cannot also occur. Therefore, the proof of Claim 5 follows.

Now, we state and prove the following.

Claim 6. There exists a positive constant $\eta _{5}$ such that $| a_{2} - a_{3} | / | a_{2} - a_{1} | \geq \eta _{5}$.

Arguing by contradiction, assume that $| a_{2} - a_{3} | = o( | a_{1} - a_{2} | )$. Multiplying $(F_{1})$ by $\frac{a_{1} - a_{2}}{ | a_{1} - a_{2} |}$, we obtain

$$\begin{aligned} & \biggl( - \overline{c}_{2} \frac{1}{\lambda _{1}} \frac{\partial \varepsilon _{12}}{\partial a_{1}} - \overline{c}_{2} \frac{1}{\lambda _{1}} \frac{\partial \varepsilon _{13}}{\partial a_{1}} \biggr) \frac{a_{1} - a_{2}}{ \vert a_{1} - a_{2} \vert } \\ &\quad = O \biggl(\varepsilon ^{2} + \frac{1}{\lambda _{1} ^{3}} \biggr) + o \biggl( \sum _{j \neq 1} \biggl\vert \frac{1}{\lambda _{1}} \frac{\partial \varepsilon _{1j}}{\partial a_{1}} \biggr\vert \biggr) +O \biggl( \frac{1}{\lambda _{1}}\sum _{j\neq 1} \varepsilon _{1j} \biggr). \end{aligned}$$

(74)

However, by Assertion (1) of Lemma 6.5, we have

$$\begin{aligned} &\biggl( - \frac{ \overline{c}_{2} }{\lambda _{1}} \frac{\partial \varepsilon _{12}}{\partial a_{1}} - \frac{ \overline{c}_{2} }{\lambda _{1}} \frac{\partial \varepsilon _{13}}{\partial a_{1}} \biggr) \frac{a_{1} - a_{2}}{ \vert a_{1} - a_{2} \vert } \\ & \quad= (n-2) \bigl( \lambda _{2} \vert a_{1}-a_{2} \vert \varepsilon _{12}^{\frac{n}{n-2}} + \lambda _{3} \vert a_{1}-a_{3} \vert \bigl(1+o(1)\bigr) \varepsilon _{13}^{\frac{n}{n-2}} \bigr) \\ & \quad\geq c \varepsilon _{12}^{(n-1)/(n-2)} + c \varepsilon _{13}^{(n-1)/(n-2)}. \end{aligned}$$

(75)

Combining (74) and (75), we derive that

$$\begin{aligned} \varepsilon _{12}; \varepsilon _{13} = o \biggl( \varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr). \end{aligned}$$

Putting this information in $(E_{1})$, we obtain (66), which gives a contradiction. Hence the proof of Claim 6 follows.

Now, we deal with the following claim.

Claim 7. There exists a positive constant $\eta _{6}$ such that $\lambda _{2} /\lambda _{3} \geq \eta _{6}$.

Arguing by contradiction, assume that $\lambda _{2} /\lambda _{3} \to 0$. Thus it follows that

$$\begin{aligned} & \varepsilon _{13} \leq \frac{1}{ (\lambda _{1} \lambda _{3} \vert a_{1} - a_{3} \vert ^{2})^{(n-2)/2}} = o \biggl( \frac{1}{ (\lambda _{1} \lambda _{2} \vert a_{2} - a_{1} \vert ^{2})^{(n-2)/2}} \biggr) = o( \varepsilon _{12}), \\ & \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{13}}{\partial a_{1}} \biggr\vert \leq \frac{ c \varepsilon _{ 13} }{\lambda _{1} \vert a_{1} - a_{3} \vert } < < \frac{ \varepsilon _{ 12} }{\lambda _{1} \vert a_{1} - a_{2} \vert } \leq \frac{c}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{1}} \biggr\vert . \end{aligned}$$

(76)

Thus $(F_{1})$ implies that

$$\begin{aligned} \varepsilon _{12}^{(n-1)/(n-2)} \leq \frac{ c \varepsilon _{12} }{ \sqrt{\lambda _{1} \lambda _{2}} \vert a_{2} - a_{1} \vert } \leq \frac{ c \varepsilon _{12} }{ \lambda _{1} \vert a_{2} - a_{1} \vert } \leq \frac{1}{\lambda _{1}} \biggl\vert \frac{\partial \varepsilon _{12}}{\partial a_{1}} \biggr\vert \leq c \biggl( \varepsilon ^{2} + \frac{1}{\lambda _{1} ^{4}} \biggr), \end{aligned}$$

which implies (68). Putting (68) and (76) in $(E_{1})$, we get (66), which gives a contradiction. Thus the proof of Claim 7 follows.

Lastly, we are going to prove the theorem in the cases mentioned above. Combining the previous claims, we get

$$\begin{aligned} & \varepsilon _{23} \leq \frac{1}{ (\lambda _{2} \lambda _{3} \vert a_{2} - a_{3} \vert ^{2})^{(n-2)/2}} = o \biggl( \frac{1}{ (\lambda _{1} \lambda _{3} \vert a_{1} - a_{3} \vert ^{2})^{(n-2)/2}} \biggr) = o( \varepsilon _{13}) \end{aligned}$$

(77)

$$\begin{aligned} & \frac{1}{\lambda _{3}} \biggl\vert \frac{\partial \varepsilon _{23}}{\partial a_{3}} \biggr\vert \leq \frac{ c \varepsilon _{ 23} }{\lambda _{3} \vert a_{2} - a_{3} \vert } < < \frac{ c \varepsilon _{ 13} }{\lambda _{3} \vert a_{1} - a_{3} \vert } \leq \frac{c}{\lambda _{3}} \biggl\vert \frac{\partial \varepsilon _{13}}{\partial a_{3}} \biggr\vert . \end{aligned}$$

(78)

Thus, using Proposition 3.6 for $i=3$, we derive that

$$\begin{aligned} \frac{1}{\lambda _{3}} \biggl\vert \frac{\partial \varepsilon _{13}}{\partial a_{3}} \biggr\vert \bigl(1+o(1) \bigr)= O (R_{3,3} + R_{4,3} + R_{5,3} + R_{6,3} ). \end{aligned}$$

Observe that, since $n\geq 6$, easy computations show that

$$\begin{aligned} R_{3,3} + R_{4,3} + R_{5,3} + R_{6,3} = o \biggl( \varepsilon ^{ \frac{n}{n-2}} +\lambda _{1}^{-\frac{2n}{n-2}}+ \frac{1}{\lambda _{3}} \biggl\vert \frac{\partial \varepsilon _{13}}{\partial a_{3}} \biggr\vert \biggr)\quad \text{and}\quad \frac{1}{\lambda _{3}} \biggl\vert \frac{\partial \varepsilon _{13}}{\partial a_{3}} \biggr\vert \geq c \varepsilon _{13}^{ n / (n-2) }. \end{aligned}$$

We derive that

$$\begin{aligned} \varepsilon _{13} = o \biggl(\varepsilon + \frac{1}{\lambda _{1} ^{2}} \biggr). \end{aligned}$$

Putting this information and (77) in $( E_{1} + E_{3} - E_{2})$, we obtain (66), which gives a contradiction. Hence the proof of the theorem is complete. □

5 Conclusion

By using delicate estimates near the “standard” bubbles, we have provided some necessary conditions to be satisfied by the concentration parameters. The careful analysis of these balancing conditions allows us to observe a new phenomenon in the higher dimensional case: the nonexistence of solutions of $(\mathcal{P}_{\varepsilon })$ that blow up at one or two or three interior points. This stands in strong contrast to the fact that if $n = 3$, then solutions to $(\mathcal{P}_{\varepsilon })$ exist with interior blow-up points [23]. However, some questions remain open:

• (i) Do the results of Theorem 1.3 remain true for all $n\geq 4$?

• (ii) Are there any solutions of $(\mathcal{P}_{\varepsilon })$ that blow up, as ε goes to zero, at N interior points with $N\geq 4$ and for all dimension $n\geq 4$?

• (iii) What happens if we put in front of the nonlinear term of $(\mathcal{P}_{\varepsilon })$ a nonconstant function K?

Appendix

In this appendix we collect several estimates needed throughout the paper. We start with the following auxiliary analysis formulae. Their proofs follow from a Taylor expansion with Lagrange remainder.

Lemma 6.1

For $1<\alpha <3$ and $\beta >0$, we have

• (1) $(a+b)^{\alpha } = a^{\alpha } +\alpha a^{\alpha -1}b + O (a^{ \alpha -2}b^{2}\chi _{b\leq a} + b^{\alpha }\chi _{a\leq b} )$,

• (2) $(a+b)^{\alpha } = a^{a} +\alpha a^{\alpha -1}b+b^{\alpha } + O ([a^{ \alpha -2}b^{2} +b^{\alpha }]\chi _{b\leq a} + [b^{\alpha -1}a +a^{ \alpha -1}b]\chi _{a\leq b} )$,

• (3) $(a+b)^{\alpha } = a^{\alpha } +b^{\alpha } + O (a^{\alpha -1}b \chi _{b\leq a} + ab^{\alpha -1}\chi _{a\leq b} )$,

• (4) $(a+b)^{\beta } = a^{\beta } +O (a^{\beta -1}b\chi _{b\leq a} + b^{ \beta }\chi _{a\leq b.} )$.

Next, we give the following estimate.

Lemma 6.2

For $\varepsilon \ln \lambda$ small and $a\in \Omega$, for each $x \in \Omega$, it holds

$$\begin{aligned} \delta _{a,\lambda }^{\varepsilon }(x)&= c_{0}^{\varepsilon } \lambda ^{ \varepsilon \frac{n-2}{2}} \biggl(1-\varepsilon \frac{n-2}{2} \ln \bigl(1+ \lambda ^{2} \vert x-a \vert ^{2}\bigr) \biggr) + O \bigl( \varepsilon ^{2} \ln ^{2} \bigl(1+ \lambda ^{2} \vert x-a \vert ^{2}\bigr) \bigr) = 1+o(1). \end{aligned}$$

Proof

By the definition of $\delta _{a,\lambda }$, we have

$$\begin{aligned} \delta _{a,\lambda }^{\varepsilon }(x)= c_{0}^{\varepsilon } \lambda ^{ \varepsilon \frac{n-2}{2}} \exp \biggl(-\varepsilon \frac{n-2}{2} \ln \bigl(1+ \lambda ^{2} \vert x-a \vert ^{2}\bigr)\biggr). \end{aligned}$$

Using the fact that Ω is bounded and Taylor’s expansion, we easily derive the desired result. □

By easy computations, we easily obtain the following result.

Lemma 6.3

For all $x\in \Omega$, it holds

$$\begin{aligned} &{{\mathrm{(i)}}}\quad \frac{1}{\lambda } \frac{\partial \delta _{a,\lambda }}{\partial a}(x) = (n-2) \frac{\lambda (x-a)}{1+\lambda ^{2} \vert x-a \vert ^{2}} \delta _{a,\lambda }(x) = \textstyle\begin{cases} O(\delta _{a,\lambda } (x) ), \\ O (\frac{\delta _{a,\lambda } (x) }{\lambda \vert x-a \vert } ), \end{cases}\displaystyle \\ &{{\mathrm{(ii)}}}\quad \lambda \frac{\partial \delta _{a,\lambda }}{\partial \lambda }(x) = \frac{n-2}{2}\delta _{a,\lambda }(x) \frac{1-\lambda ^{2} \vert x-a \vert ^{2}}{1+\lambda ^{2} \vert x-a \vert ^{2}}= O \bigl( \delta _{a,\lambda }(x) \bigr). \end{aligned}$$

Next, we give the following estimates.

Lemma 6.4

We have

$$\begin{aligned} & (1)\quad \int _{\Omega }(\delta _{i}\delta _{j})^{n/(n-2)} \leq C \varepsilon _{ij}^{n/(n-2)}\ln \varepsilon _{ij}^{-1}, \\ &(2)\quad \int _{\Omega }\delta _{i} ^{\frac{2n}{n-2}-\beta }\ln ^{ \gamma }\bigl(1+\lambda ^{2} \vert x-a \vert ^{2} \bigr) \leq \frac{C}{\lambda _{i}^{\beta \frac{n-2}{2}}}\quad \forall \beta \in [0,\frac{n}{n-2}) \quad \forall \gamma \geq 0, \\ &(3)\quad \int _{\mathbb{R}^{n}\setminus B(a,r)}\delta _{a,\lambda }^{(2n)/(n-2)} \leq \frac{C}{(\lambda r)^{n}}, \\ &(4)\quad \int _{\mathbb{R}^{n}}\delta _{i}^{\frac{n+2}{n-2}} \frac{1}{\lambda _{j}}\frac{\partial \delta _{j}}{\partial a_{j}}= \frac{n+2}{n-2} \int _{\mathbb{R}^{n}}\delta _{j}^{\frac{4}{n-2}} \frac{1}{\lambda _{j}}\frac{\partial \delta _{j}}{\partial a_{j}} \delta _{i}= \overline{c}_{2}\frac{1}{\lambda _{j}} \frac{\partial \varepsilon _{ij}}{\partial a_{j}} +O \bigl(\lambda _{i} \vert a_{i}-a_{j} \vert \varepsilon _{ij}^{\frac{n+1}{n-2}} \bigr), \\ &(5)\quad \int _{\mathbb{R}^{n}}\delta _{j}^{\frac{n+2}{n-2}}\lambda _{i} \frac{\partial \delta _{i}}{\partial \lambda _{i}}= \frac{n+2}{n-2} \int _{\mathbb{R}^{n}}\delta _{j} \delta _{i}^{\frac{4}{n-2}} \lambda _{i} \frac{\partial \delta _{i}}{\partial \lambda _{i}}= \overline{c}_{2} \lambda _{i}\frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}} +O \bigl(\varepsilon _{ij}^{\frac{n}{n-2}} \ln \varepsilon _{ij}^{-1} \bigr), \end{aligned}$$

where $\overline{c}_{2}:= \int _{ \mathbb{R}^{n}} \frac{c_{0} ^{2n/(n-2)} }{(1+ | x |^{2})^{(n+2)/2}}$.

Proof

(1), (4), and (5) are extracted from estimates E2 (page 4), F11 (page 22), and F16 (page 23) of [5] respectively. However, (2) and (3) follow by using standard computations. □

Now, we state the following properties.

Lemma 6.5

We have

$$\begin{aligned} & (1) \quad\frac{1}{\lambda _{j}} \frac{\partial \varepsilon _{ij}}{\partial a_{j}}= (n-2)\lambda _{i}(a_{i}-a_{j}) \varepsilon _{ij}^{n/(n-2)} \quad\textit{and}\quad {\lambda _{j}} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{j}} = - \frac{n-2}{2} \varepsilon _{ij} \biggl( 1 - 2 \frac{\lambda _{i}}{\lambda _{j}} \varepsilon _{ij}^{2/(n-2)} \biggr), \\ & (2)\quad -\lambda _{j} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{j}} +\lambda _{i} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}} \geq 0 \quad\textit{if } \lambda _{i}\leq \lambda _{j}, \\ &(3)\quad -2\lambda _{j} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{j}} -\lambda _{i} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}}\geq c \varepsilon _{ij} \quad\textit{if } \lambda _{i}\leq \lambda _{j} \quad\textit{and}\quad -\lambda _{j} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{j}} -\lambda _{i} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}}\geq 0, \\ &(4)\quad \frac{c}{(\lambda _{i}\lambda _{j} \vert a_{i}-a_{j} \vert ^{2})^{\frac{n-2}{2}}} \leq \varepsilon _{ij}\leq \frac{1}{(\lambda _{i}\lambda _{j} \vert a_{i}-a_{j} \vert ^{2})^{\frac{n-2}{2}}}\quad \textit{if } \lambda _{i}\leq \lambda _{j} \quad\textit{and}\quad \lambda _{i} \vert a_{i}-a_{j} \vert \geq C, \\ &(5)\quad c \biggl(\frac{\lambda _{i}}{\lambda _{j}} \biggr)^{ \frac{n-2}{2}}\leq \varepsilon _{ij}\leq \biggl( \frac{\lambda _{i}}{\lambda _{j}} \biggr)^{\frac{n-2}{2}}\quad \textit{if } \lambda _{i}\leq \lambda _{j} \textit{ and } \lambda _{i} \vert a_{i}-a_{j} \vert \leq C, \\ &(6)\quad - \lambda _{j} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{j}}= \frac{n-2}{2} \varepsilon _{ij} + O \bigl(\varepsilon _{ij}^{n/(n-2)} \bigr) \quad\textit{if } \lambda _{i}/\lambda _{j} \not \to \infty, \\ &(7) \quad\varepsilon _{ij}= \biggl(\frac{\lambda _{i}}{\lambda _{j}} \biggr)^{\frac{n-2}{2}}\bigl(1+o(1)\bigr) \quad\textit{and}\quad \lambda _{i} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}}= \frac{n-2}{2} \varepsilon _{ij}\bigl(1+o(1) \bigr)\quad \textit{if } \lambda _{i}\leq \lambda _{j} \\ & \qquad\textit{and } \lambda _{i} \vert a_{i}-a_{j} \vert \to 0, \\ &(8)\quad \varepsilon _{ij}= \frac{1+o(1)}{(\lambda _{i}\lambda _{j} \vert a_{i}-a_{j} \vert ^{2})^{(n-2)/2}} \quad\textit{if } \lambda _{i} \textit{ and } \lambda _{j} \textit{ are of the same order}, \\ &(9)\quad - \lambda _{j} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{j}}= \frac{n-2}{2} \varepsilon _{ij} \bigl(1+o(1)\bigr) \quad\textit{if } \lambda _{j} \vert a_{i}-a_{j} \vert \to \infty. \end{aligned}$$

Proof

Claim (1) follows immediately from the definition of $\varepsilon _{ij}$ (see (4)). Concerning Claim (2), using the second assertion of Claim (1), we get, for $\lambda _{i}\leq \lambda _{j}$,

$$\begin{aligned} -\lambda _{j}\frac{\partial \varepsilon _{ij}}{\partial \lambda _{j}} + \lambda _{i} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}}= (n-2)\varepsilon _{ij}^{n/(n-2)} \biggl( \frac{\lambda _{j}}{\lambda _{i}} - \frac{\lambda _{i}}{\lambda _{g}} \biggr) \geq 0, \end{aligned}$$

which completes the proof of Claim (2). In the same way, we have

$$\begin{aligned} -\lambda _{j}\frac{\partial \varepsilon _{ij}}{\partial \lambda _{j}} - \lambda _{i} \frac{\partial \varepsilon _{ij}}{\partial \lambda _{i}}= (n-2)\varepsilon _{ij}^{n/(n-2)} \bigl( \lambda _{i}\lambda _{j} \vert a_{i}-a_{j} \vert ^{2} \bigr), \end{aligned}$$

which implies the second assertion of Claim (3). Furthermore, for $\lambda _{i}\leq \lambda _{j}$, we have

$$\begin{aligned} -\lambda _{j}\frac{\partial \varepsilon _{ij}}{\partial \lambda _{j}} = \frac{n-2}{2}\varepsilon _{ij} \biggl(1- \frac{\lambda _{i}}{\lambda _{j}}\varepsilon _{ij}^{2/(n-2)} \biggr)= \frac{n-2}{2}\varepsilon _{ij}\bigl(1+o(1)\bigr), \end{aligned}$$

which completes the proof of Claim (3).

Now, assuming that $\lambda _{i}\leq \lambda _{j}$ and $\lambda _{i}|a_{i}-a_{j}|\geq C$, we see that

$$\begin{aligned} 1\leq \frac{\varepsilon _{ij}^{-2/(n-2)}}{\lambda _{i}\lambda _{j} \vert a_{i}-a_{j} \vert ^{2}}= \frac{1}{\lambda _{j}^{2} \vert a_{i}-a_{j} \vert ^{2}} + \frac{1}{\lambda _{i}^{2} \vert a_{i}-a_{j} \vert ^{2}} + 1 \leq c. \end{aligned}$$

Hence the proof of Claim (4) is completed.

Concerning Claim (5), we observe that if $\lambda _{i}\leq \lambda _{j}$ and $\lambda _{i}|a_{i}-a_{j}|\leq C$, we have

$$\begin{aligned} 1\leq \frac{\lambda _{i}}{\lambda _{j}}\varepsilon _{ij}^{-2/(n-2)}= 1 + \frac{\lambda _{i}^{2}}{\lambda _{j}^{2}}+ \lambda _{i}^{2} \vert a_{i}-a_{j} \vert ^{2} \leq c. \end{aligned}$$

Thus Claim (5) follows.

Notice that Claim (6) follows immediately from the second assertion of Claim (1).

Concerning Claim (7), observe that if $\lambda _{i}\leq \lambda _{j}$ and $\lambda _{i}|a_{i}-a_{j}|\to 0$, then from the smallness of $\varepsilon _{ij}$ it follows that $\lambda _{i}/\lambda _{j}$ is very small. Thus

$$\begin{aligned} \lambda _{i}\lambda _{j} \vert a_{i}-a_{j} \vert ^{2}= \bigl(\lambda _{i}^{2} \vert a_{i}-a_{j} \vert ^{2} \bigr) \frac{\lambda _{j}}{\lambda _{i}}=o \biggl( \frac{\lambda _{j}}{\lambda _{i}} \biggr). \end{aligned}$$

Hence Claim (7) follows.

Now, note that if $\lambda _{i}$ and $\lambda _{j}$ are of the same order, then the smallness of $\varepsilon _{ij}$ implies that $\lambda _{i}\lambda _{j}|a_{i}-a_{j}|^{2}$ is very large. Hence Claim (8) follows.

Finally, if $\lambda _{i}|a_{i}-a_{j}|\to \infty$, then we have

$$\begin{aligned} \frac{\lambda _{i}}{\lambda _{j}}\varepsilon _{ij}^{2/(n-2)} = \frac{1}{1+\frac{\lambda _{j}^{2}}{\lambda _{i}^{2}}+\lambda _{j}^{2} \vert a_{i}-a_{j} \vert ^{2}} \to 0, \end{aligned}$$

which implies Claim (9), and therefore the proof of the lemma is completed. □

We end this appendix by proving the following two useful results.

Lemma 6.6

Let $n \geq 3$.

(1) For $1\leq \beta < n/(n-2)$ and $\lambda _{i} \leq \lambda _{j}$, it holds

$$\begin{aligned} \int _{\Omega }(\delta _{i}\delta _{j})^{\beta } \leq C \varepsilon _{ij}^{ \beta } \textstyle\begin{cases} \frac{ 1 }{ \lambda _{i} ^{\beta (n-2) } } + \vert a_{i} - a_{j} \vert ^{ \beta (n-2) } & \textit{if } 2 \beta ( n-2 ) < n, \\ \vert a_{i} - a_{j} \vert ^{ \beta (n-2) } \vert \ln \vert a_{i} - a_{j} \vert \vert + \frac{1}{\lambda _{i} ^{ \beta (n-2) }} \ln \lambda _{i} & \textit{if } 2 \beta ( n-2 ) = n, \\ { \vert a_{i} - a_{j} \vert ^{ n - \beta (n-2 ) } } + \frac{1}{\lambda _{i} ^{ n - \beta (n-2) }} & \textit{if } 2 \beta ( n-2 ) > n. \end{cases}\displaystyle \end{aligned}$$

(2) Let $1 \leq \alpha$ and $1 \leq \beta$ be such that $\alpha \neq \beta$ and (i): $\alpha + \beta = 2n/(n-2)$ or (ii): $\alpha + \beta < 2n/(n-2)$. Then it holds

$$\begin{aligned} \int _{\Omega }\delta _{i} ^{\alpha }\delta _{j} ^{\beta }\leq C \varepsilon _{ij}^{ \min (\alpha, \beta ) };\qquad \int _{ \Omega }\delta _{i} ^{\alpha }\delta _{j} ^{\beta }= o\bigl( \varepsilon _{ij}^{ \min (\alpha, \beta ) } \bigr)\quad \textit{in case } (ii). \end{aligned}$$

Proof

We remark that, for $\beta = n / (n-2)$, the estimate of Claim (1) is already given in Lemma 6.4. Here, we need to improve this estimate when $\beta < n /(n-2)$. Furthermore, the case $2 \beta (n-2) < n$ occurs only when $n = 3$ and $\beta < 3/2$, and the case $2 \beta (n-2) = n$ occurs only if $n = 4$ and $\beta = 1$ or $n = 3$ and $\beta = 3/2$.

First, we focus on proving Claim (1).

Note that, if $2 \beta (n-2) < n$ (this case can occur only if $n = 3$ and $\beta < 3/2$), in this case, it holds

$$\begin{aligned} \int _{\Omega }(\delta _{i}\delta _{j})^{\beta } \leq \frac { c }{ ( \lambda _{i} \lambda _{j} )^{ \beta (n-2)/2}} \biggl( \sum_{k=i,j} \int _{\Omega } \frac{dx}{ \vert x - a_{k} \vert ^{ 2 \beta (n-2) }} \biggr)\leq c \varepsilon _{ij}^{ \beta } \biggl( \frac{ 1 }{ \lambda _{i} ^{\beta (n-2) } } + \vert a_{i} - a_{j} \vert ^{ \beta (n-2) } \biggr). \end{aligned}$$

Thus, in the sequel of the proof, we consider $2 \beta (n-2) \geq n$. We distinguish two cases.

Case 1. $\lambda _{i}|a_{i}-a_{j}|\leq M$, where M is a large constant. In this case, by Lemma 6.5, we know that $\varepsilon _{ij}$ and $({\lambda _{i}} / {\lambda _{j}} )^{ ( n-2 ) / {2}}$ have the same order. Let $B_{j}:= B(a_{j}, 4 M / {\lambda _{i}} )$. Observe that, for $x \in \Omega \setminus B_{j}$, we have $| x-a_{i} | \geq c | x - a_{j} |$. Thus

$$\begin{aligned} \int _{\Omega }(\delta _{i}\delta _{j})^{\beta }& \leq C \biggl( \frac{\lambda _{i}}{\lambda _{j}} \biggr)^{\beta \frac{n-2}{2}} \int _{B_{j}} \frac{dx}{ \vert x - a_{j} \vert ^{\beta (n-2)}} + \frac{ c }{ ( \lambda _{i}\lambda _{j} ) ^{\beta \frac{n-2}{2}}} \int _{ \Omega \setminus B_{j}} \frac{dx}{ \vert x - a_{j} \vert ^{ 2 \beta (n-2)}} \\ & \leq C \varepsilon _{ij}^{\beta } \frac{1}{\lambda _{i} ^{ n - \beta (n-2) }} + C \varepsilon _{ij}^{ \beta }\frac{ 1 }{ \lambda _{i} ^{\beta ( n-2 ) }} \textstyle\begin{cases} \ln \lambda _{i} & \text{if } 2 \beta ( n-2 ) = n, \\ { \lambda _{i} ^{2 \beta (n-2 ) - n} } & \text{if } 2 \beta ( n-2 ) > n . \end{cases}\displaystyle \end{aligned}$$

Hence the result in this case.

Case 2. $\lambda _{i}|a_{i}-a_{j}|\geq M$. In this case, Lemma 6.5 implies that $(\lambda _{i}\lambda _{j}|a_{i}-a_{j}|^{2} )^{(2 - n) / {2}}$ and $\varepsilon _{ij}$ have the same order. For $k=i, j$, let $B_{k}=B(a_{k}, |a_{i}-a_{j}|/4)$. Observe that

$$\begin{aligned} \int _{\Omega }(\delta _{i}\delta _{j})^{\beta } \leq {}&\frac{1}{ (\lambda _{i}\lambda _{j} \vert a_{i}-a_{j} \vert ^{2} )^{{\beta (n-2)} / {2}}} \sum_{k\in \{i,j\}} \int _{B_{k}} \frac{dx}{ \vert x-a_{k} \vert ^{\beta (n-2)}}+ \biggl( \int _{\Omega \setminus B_{i}}\delta _{i}^{2\beta } \biggr)^{ \frac{1}{2}} \biggl( \int _{\Omega \setminus B_{j}}\delta _{j}^{2\beta } \biggr)^{ \frac{1}{2 }} \\ \leq{}& C \varepsilon _{ij}^{\beta } \vert a_{i} - a_{j} \vert ^{ n - \beta (n-2) } + \prod_{k=i,j} \biggl( \frac{ 1 }{ \lambda _{k} ^{ \beta (n-2) } } \int _{\Omega \setminus B_{k}} \frac{ 1 }{ \vert x - a_{k} \vert ^{ 2 \beta (n-2) } } \biggr)^{{1} / {2}} \\ \leq{}& C \varepsilon _{ij}^{\beta } \vert a_{i} - a_{j} \vert ^{ n - \beta (n-2) } \\ &{} + C \varepsilon _{ij}^{\beta } \vert a_{i} - a_{j} \vert ^{ \beta (n-2) } \textstyle\begin{cases} \vert \ln \vert a_{i} - a_{j} \vert \vert & \text{if } 2 \beta ( n-2 ) = n, \\ { \vert a_{i} - a_{j} \vert ^{ n - 2 \beta (n-2 ) } } & \text{if } 2 \beta ( n-2 ) > n. \end{cases}\displaystyle \end{aligned}$$

Hence the result follows in this case also.

Hence the proof of Claim (1) is complete. Concerning Claim (2), it follows from Claim $(d)$ of Lemma 2.2 of [8] when the assumption (i) is satisfied. However, when (ii) is satisfied (assume that $\alpha < \beta$), let $\gamma:= (\alpha + \beta ) / 2$, it follows that $\gamma < n / (n-2)$ and $\gamma - \alpha = ( \beta - \alpha ) / 2$. Thus, using Holder’s inequality and Claim (1), it holds

$$\begin{aligned} \int _{\Omega }\delta _{i} ^{\alpha }\delta _{j} ^{\beta }= \int _{ \Omega }( \delta _{i} \delta _{j} ) ^{\alpha }\delta _{j} ^{\beta - \alpha } \leq \biggl( \int _{\Omega }( \delta _{i} \delta _{j} ) ^{ \gamma } \biggr)^{ \alpha / \gamma } \biggl( \int _{\Omega }\delta _{j} ^{ \gamma / 2 } \biggr)^{ ( \gamma - \alpha ) / \gamma } = o \bigl( \varepsilon _{ij} ^{\alpha } \bigr). \end{aligned}$$

Hence the proof is complete. □

Lemma 6.7

For $a \in \Omega$ and $\beta > 0$, it holds

$$\begin{aligned} \int _{\partial \Omega } \frac{ 1 }{ \vert a - y \vert ^{ n-1+ \beta } } \,dy \leq \frac{ c }{ d (a,\partial \Omega ) ^{\beta }}. \end{aligned}$$

Proof

We remark that if a is far away from the boundary, then the result is immediate. Hence, we focus on the case where $d_{a}:= d(a,\partial \Omega )$ is small. Let a̅ be the projection of a at the boundary. Thus, we have

$$\begin{aligned} \int _{ \partial \Omega } \frac{ 1 }{ \vert a - y \vert ^{ n - 1 +\beta } } \,dy & \leq c \int _{ \partial \Omega } \frac{ 1 }{ (d_{a} ^{2} + \vert \overline{a} - y \vert ^{2})^{ (n - 1 + \beta )/2 } } \,dy \\ & \leq \frac{ c }{ d_{a} ^{n - 1 + \beta } } \int _{ \partial \Omega \cap B(\overline{a},1)} \frac{ 1 }{ (1 + d_{a} ^{-2} \vert \overline{a} - y \vert ^{2})^{ (n + 1)/2 } } \,dy + c \\ & \leq \frac{ c }{ d_{a} ^{ \beta } } \int _{ \mathbb{R}^{n-1}} \frac{ 1 }{ (1 + \vert y \vert ^{2})^{ (n - 1 + \beta ) / 2 } } \,dy + c \leq \frac{ c }{ d_{a} ^{ \beta } }. \end{aligned}$$

 □