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Multiplicity results for biharmonic equations involving multiple Rellich-type potentials and critical exponents

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Abstract

In this paper, a biharmonic equation is investigated, which involves multiple Rellich-type potentials and a critical Sobolev exponent. By using variational methods and analytical techniques, the existence and multiplicity of nontrivial solutions to the equation are established.

Introduction

In this paper, we study the following biharmonic equation:

figurea

where \(\varOmega \subset \mathbb{R}^{N}\) (\(N\ge 5\)) is a smooth bounded domain such that the different points \(a_{i}\in \varOmega \), \(i=1,2,\ldots ,k\), \(k\ge 2\), \(\frac{\partial }{\partial n}\) is the outward normal derivative, \(0\le \mu _{i}<\bar{\mu }:=(\frac{N(N-4)}{4})^{2}\), \(\lambda >0\), \(1\le q<2^{*}\), and \(2^{*}:=\frac{2N}{N-4}\) is the critical Sobolev exponent.

Equation (\(E_{\lambda }\)) is related to the following Rellich inequality [22]:

$$\begin{aligned} \int _{\varOmega }\frac{u^{2}}{ \vert x-a \vert ^{4}} \,dx\le \frac{1}{\bar{\mu }} \int _{\varOmega } \vert \Delta u \vert ^{2} \,dx, \quad \forall a\in \varOmega , u \in H^{2}_{0}(\varOmega ), \end{aligned}$$
(1.1)

where \(H_{0}^{2}(\varOmega )\) is the completion of \(C_{0}^{\infty }( \varOmega )\) with respect to \((\int _{\varOmega }|\Delta \cdot |^{2}\,dx)^{1/2}\). Then the following best constant is well defined:

$$\begin{aligned} A_{\mu }(\varOmega ):=\inf_{ H_{0}^{2}(\varOmega )\setminus \{0\}}\frac{ \int _{\varOmega }( \vert \Delta u \vert ^{2}-\mu \frac{u^{2}}{ \vert x-a \vert ^{4}})\,dx}{( \int _{\varOmega } \vert u \vert ^{2^{*}} \,dx)^{\frac{2}{2^{*}}}}, \quad \forall a\in \varOmega , \mu < \bar{\mu }. \end{aligned}$$

Note that it is well known that \(A_{\mu }(\varOmega )\) is independent of Ω and that \(A_{\mu }(\varOmega )\) is not obtained except in the case with \(\varOmega =\mathbb{R}^{N}\). Moreover, the minimizers of \(A_{\mu }(\varOmega )\) have been investigated by some authors (e.g. [3, 10, 11, 19]). Thus, we will simply denote \(A_{\mu }( \varOmega )=A_{\mu }(\mathbb{R}^{N})=A_{\mu }\).

In this paper, for \(\sum_{i=1}^{k}\mu _{i}\in [0,\bar{\mu })\), we use \(H_{0}^{2}(\varOmega )\) to denote the completion of \(C_{0}^{\infty }( \varOmega )\) with respect to the norm

$$\begin{aligned} \Vert u \Vert := \Biggl( \int _{\varOmega }\Biggl( \vert \Delta u \vert ^{2}-\sum _{i=1}^{k}\frac{\mu _{i}u ^{2}}{ \vert x-a_{i} \vert ^{4}}\Biggr) \,dx \Biggr)^{\frac{1}{2}}. \end{aligned}$$

By (1.1), this norm is equivalent to the usual norm \((\int _{ \varOmega }|\Delta u|^{2}\,dx)^{\frac{1}{2}}\).

It is easily to see that Eq. (\(E_{\lambda }\)) is variational and its solutions are critical points of the functional defined in \(H_{0}^{2}(\varOmega )\) by

$$\begin{aligned} J_{\lambda }(u):=\frac{1}{2} \Vert u \Vert ^{2}- \frac{1}{2^{*}} \int _{\varOmega } \vert u \vert ^{2^{*}}-\frac{ \lambda }{q} \int _{\varOmega } \vert u \vert ^{q},\quad u\in H_{0}^{2}(\varOmega ). \end{aligned}$$

Then \(J_{\lambda }\in C^{1}(H_{0}^{2}(\varOmega ),\mathbb{R})\) and that

$$\begin{aligned} \bigl\langle J_{\lambda }'(u),v\bigr\rangle = \int _{\varOmega }\Biggl(\Delta u\Delta v- \sum _{i=1}^{k}\frac{\mu _{i} uv}{ \vert x-a_{i} \vert ^{4}}\Biggr)- \int _{\varOmega } \vert u \vert ^{2^{*}-2} uv- \lambda \int _{\varOmega } \vert u \vert ^{q-2}uv,\quad \forall v \in H_{0}^{2}(\varOmega ). \end{aligned}$$

In recent years problems related with the inequality (1.1) and the equations with biharmonic operator have been investigated in several works; we quote [1, 3, 6,7,8,9,10, 13, 18, 19]. On the other hand, the biharmonic problems involving a Rellich-type potential and a critical Sobolev exponent have seldom been studied; we only find some results in [10, 18, 19]. Thus it is necessary for us to investigate the related biharmonic problems deeply. Very recently, Hsu and Zhang [16] studied the existence and multiplicity of nontrivial solution for the following equation:

$$ \textstyle\begin{cases} \Delta ^{2} u -\frac{\mu }{ \vert x \vert ^{4}}u =\frac{ \vert u \vert ^{2^{*}(s)-2}}{ \vert x \vert ^{s}}u+ \lambda \frac{ \vert u \vert ^{q-2}}{ \vert x \vert ^{t}}u, &x\in \varOmega , \\ u =\frac{\partial u}{\partial n}=0, & x\in \partial \varOmega , \end{cases} $$

where \(\varOmega \subset \mathbb{R}^{N}\) (\(N\ge 5\)) is a smooth bounded domain such that \(0\in \varOmega \), \(0\le \mu <\bar{\mu }\), \(0\le s\), \(t<4\), \(1\le q<2\), \(\lambda >0\).

In this paper, we study a biharmonic equation involving multiple Rellich-type potentials and a critical Sobolev exponent. It should be mentioned that the main technical difficulty to study equations like Eq. (\(E_{\lambda }\)) is the lack of knowledge of the explicit form minimizers to the best Rellich–Sobolev constant \(A_{\mu _{i}}\). However, as in [10] and [19], this difficulty can be overcome since the unique tool which is necessary to perform the needed asymptotic expansions is the asymptotic behavior at the origin and infinity of Rellich–Sobolev extremals and their first derivatives, which is established in Theorem 1.1 of [19]. We are only aware of the work in [18] which studied the existence and nonexistence of ground state solution to Eq. (\(E_{\lambda }\)) when \(\varOmega =\mathbb{R}^{N}\), \(k\ge 2\) and \(\lambda =0\). Furthermore, Eq. (\(E_{\lambda }\)) have never been studied when Ω is a smooth bounded domain and \(k\ge 2\), and our results are new.

For \(0\le \mu _{i}<\bar{\mu }\) and \(a_{i}\in \varOmega \), \(i=1,2,\ldots , k\), we can define the constant:

$$\begin{aligned} A_{\mu _{i}}:=\inf_{u\in H_{0}^{2}(\varOmega )\setminus \{0\}}\frac{ \int _{\varOmega }( \vert \Delta u \vert ^{2}-\mu _{i}\frac{u^{2}}{ \vert x-a_{i} \vert ^{4}})\,dx}{( \int _{\varOmega } \vert u \vert ^{2^{*}}\,dx)^{\frac{2}{2^{*}}}}. \end{aligned}$$

The authors in [10, 19] proved that \(A_{\mu _{i}}\) is attained in \(\mathbb{R}^{N}\) by the functions

$$\begin{aligned} \bigl\{ y_{\varepsilon }^{\mu _{i}}(x-a_{i})=\varepsilon ^{\frac{4-N}{2}} U _{\mu _{i}}\bigl({\varepsilon }^{-1} (x-a_{i})\bigr), \varepsilon >0\bigr\} , \end{aligned}$$
(1.2)

where \(U_{\mu _{i}}(x)\) is positive, radially symmetric, radially decreasing, and solves

$$\Delta ^{2} u-\mu _{i} \frac{u}{ \vert x \vert ^{4}}= \vert u \vert ^{2^{*}-1}, \quad x\in \mathbb{R}^{N}\setminus \{0\}, u>0, $$

which satisfies

$$\int _{\mathbb{R}^{N}} \biggl( \bigl\vert \Delta y_{\varepsilon }^{\mu _{i}}(x-a_{i}) \bigr\vert ^{2}- \mu _{i} \frac{ \vert y_{\varepsilon }^{\mu _{i}}(x-a_{i}) \vert ^{2}}{ \vert x-a_{i} \vert ^{4}} \biggr)\,dx= \int _{\mathbb{R}^{N}} \bigl\vert y_{\varepsilon }^{\mu _{i}}(x-a_{i}) \bigr\vert ^{2^{*}}\,dx=A _{\mu _{i}}^{\frac{N}{4}}. $$

Moreover, by setting \(\rho =|x|\),

$$\begin{aligned} & U_{\mu }(\rho )=O_{1}\bigl(\rho ^{-a(\mu )}\bigr),\quad \text{as }\rho \to 0, \\ & U_{\mu }(\rho )=O_{1}\bigl(\rho ^{-b(\mu )}\bigr),\qquad U_{\mu }'(\rho )=O _{1}\bigl(\rho ^{-b(\mu )-1}\bigr),\quad \text{as }\rho \to +\infty , \end{aligned}$$

where \(a(\mu ):=\frac{N-4}{2}f(\mu )\), \(b(\mu ):=\frac{N-4}{2}(2-f( \mu ))\) and \(f:[0,\bar{\mu }]\to [0,1]\) is defined as

$$f(\mu ):=1-\frac{\sqrt{N^{2}-4N+8-4\sqrt{(N-2)^{2}+\mu }}}{N-4}, \quad \mu \in [0,\bar{\mu }]. $$

From Lemma 2.1 in [18], it follows that for \(\mu \in [0,\bar{\mu })\)

$$\begin{aligned} 0\le a(\mu ) \le \delta \le b(\mu ) \le 2\delta , \quad \delta := \frac{ N-4}{2}. \end{aligned}$$
(1.3)

Furthermore, there exist positive constants \(\mathcal{C}_{1}(\mu )\) and \(\mathcal{C}_{2}(\mu )\) such that

$$0< \mathcal{C}_{1}(\mu )\le U_{\mu }(x) \bigl( \vert x \vert ^{\frac{a(\mu )}{\delta }}+ \vert x \vert ^{\frac{b( \mu )}{\delta }}\bigr)^{\delta }\le \mathcal{C}_{2}(\mu ),\quad \forall x \in \mathbb{R}^{N} \setminus \{0\}. $$

Without loss of generality, throughout this paper we assume that

\((\mathcal{H})\) :

\(0\le \mu _{1}\le \mu _{2}\le \cdots \le \mu _{k}<\bar{ \mu }\), \(\sum_{i=1}^{k}\mu _{i}<\bar{\mu }\), and \(2^{*}:= \frac{2N}{N-4}\).

In this paper, we define the following constants and notations:

$$\begin{aligned} & \Vert u \Vert ^{2}= \int _{\varOmega }\Biggl( \vert \Delta u \vert ^{2}-\sum _{i=1}^{k}\frac{\mu _{i}u ^{2}}{ \vert x-a_{i} \vert ^{4}}\Biggr) \,dx \text{ is the norm in } H_{0}^{2}(\varOmega ); \\ &H^{-2}(\varOmega ):\text{ the dual space of } H^{2}_{0}( \varOmega ); \\ & \langle \cdot ,\cdot \rangle :\text{ the usual scalar product in } H_{0}^{2}(\varOmega ); \\ &B_{r}(a)=\bigl\{ x: \vert x-a \vert < r\bigr\} , \qquad \overline{B_{r}(a})=\bigl\{ x: \vert x-a \vert \le r\bigr\} , \quad a \in \mathbb{R^{N}},\quad r>0 ; \\ & \mu ^{*}:=\frac{1}{16}\bigl(N^{2}-16\bigr) \bigl(N^{2}-8N\bigr),\quad N\ge 9; \\ & S=\inf_{u\in H_{0}^{2}(\varOmega )\setminus \{0\}}\frac{\int _{\varOmega }( \vert \Delta u \vert ^{2}-\sum_{i=1}^{k}\mu _{i}\frac{u^{2}}{ \vert x-a_{i} \vert ^{4}})\,dx}{ (\int _{\varOmega } \vert u \vert ^{2^{*}}\,dx )^{\frac{2}{2^{*}}}}; \end{aligned}$$
(1.4)
$$\begin{aligned} & \varLambda _{0}:= \biggl(\frac{2-q}{2^{*}-q} \biggr)^{\frac{2-q}{2^{*}-2}} \biggl(\frac{2^{*}-2}{2^{*}-q} \biggr) \vert \varOmega \vert ^{- \frac{2^{*}-q}{2^{*}}}S^{\frac{(2-q)N}{8}+\frac{q}{2}}; \end{aligned}$$
(1.5)
$$\begin{aligned} &\lambda _{1}:=\inf_{u\in H_{0}^{2}(\varOmega )\setminus \{0\}}\frac{ \int _{\varOmega }( \vert \Delta u \vert ^{2}-\sum_{i=1}^{k}\mu _{i}\frac{u^{2}}{ \vert x-a _{i} \vert ^{4}})\,dx}{\int _{\varOmega } \vert u \vert ^{2}\,dx}. \end{aligned}$$
(1.6)

Since the embedding \(H_{0}^{2}(\varOmega )\hookrightarrow L^{2}(\varOmega )\) is compact, by choosing a minimizing sequence, we easily infer that \(\lambda _{1}\) can be obtained in \(H_{0}^{2}(\varOmega )\), and \(\lambda _{1}>0\). \(C, C_{1}, C_{2},\ldots \) denote various positive constants. For all \(\varepsilon >0\), \(\tau >0\), \(O(\varepsilon ^{\tau })\) denotes the quantity satisfying \(|O(\varepsilon ^{\tau })/\varepsilon ^{\tau }| \le C\) and \(o(\varepsilon ^{\tau })\) means \(|o(\varepsilon ^{\tau })/ \varepsilon ^{\tau }|\to 0\) as \(\varepsilon \to \varepsilon _{0}\), \(o_{n}(1)\) denotes \(o_{n}(1)\to 0\) as \(n\to \infty \) and \(O_{1}( \varepsilon ^{\tau })\) \((\varepsilon \to \varepsilon _{0})\) means that there exist the constants \(C_{1}\), \(C_{2}>0\) such that \(C_{1} \varepsilon ^{\tau }\le O_{1}(\varepsilon ^{\tau })\le C_{2} \varepsilon ^{\tau }\) as \(\varepsilon \to \varepsilon _{0}\). \(|\varOmega |\) denotes the Lebesgue measure of Ω and omit dx in integrals for convenience.

Let \(1\le q<2^{*}\), by the Hölder inequality and (1.4), for all \(u\in H^{2}_{0}(\varOmega )\), we obtain

$$ \begin{aligned} \int _{\varOmega } \vert u \vert ^{q}\le \biggl( \int _{\varOmega } 1 \biggr)^{ \frac{2^{*}-q}{2^{*}}} \biggl( \int _{\varOmega } \vert u \vert ^{2^{*}} \biggr)^{ \frac{q}{2^{*}}} \le \vert \varOmega \vert ^{\frac{2^{*}-q}{2^{*}}}S^{-\frac{q}{2}} \Vert u \Vert ^{q}. \end{aligned} $$
(1.7)

We are now ready to state our main results.

Theorem 1.1

Let \(N\ge 5\), \(1\le q<2\) and assume that \((\mathcal{H})\) holds, then we have the following results.

  1. (i)

    If \(\lambda \in (0,\varLambda _{0})\), then Eq. (\(E_{\lambda }\)) has at least one nontrivial solution.

  2. (ii)

    If \(\lambda \in (0,\frac{q}{2}\varLambda _{0})\), then Eq. (\(E_{\lambda }\)) has at least two nontrivial solutions.

Theorem 1.2

Let \(N\ge 5\), \(2\le q<2^{*}\) and assume that \((\mathcal{H})\) and one of the following conditions holds:

  1. (i)

    \(\lambda >0\), \(\overline{q}< q<2^{*}\), where

    $$\overline{q}=\max \biggl\{ 2, \frac{N}{b(\mu _{k})}, \frac{4(N-2-b( \mu _{k}))}{N-4} \biggr\} . $$
  2. (ii)

    \(N\ge 8\), \(0<\lambda <\lambda _{1}\), \(q=2\), \(0\le \mu _{k} \le \mu ^{*}\).

Then Eq. (\(E_{\lambda }\)) has at least one nontrivial solution.

This paper is organized as follows. In Sect. 2, we give some properties of Nehari manifold. In Sects. 3 and 4, we prove Theorem 1.1. In Sect. 5, we prove Theorem 1.2.

Nehari manifold

In this section, we will give some properties of Nehari manifold. As the energy functional \(J_{\lambda }\) is not bounded below on \(H^{2}_{0}( \varOmega )\), it is useful to consider the functional on the Nehari manifold

$$\mathcal{M}_{\lambda }=\bigl\{ u\in H^{2}_{0}(\varOmega )\setminus \{0\}: \bigl\langle J_{\lambda }'(u),u\bigr\rangle =0\bigr\} . $$

Thus, \(u\in \mathcal{M}_{\lambda }\) if and only if

$$\begin{aligned} \bigl\langle J_{\lambda }'( u),u\bigr\rangle = \Vert u \Vert ^{2}- \int _{\varOmega } \vert u \vert ^{2^{*}}- \lambda \int _{\varOmega } \vert u \vert ^{q}=0. \end{aligned}$$
(2.1)

Note that \(\mathcal{M}_{\lambda }\) contains every nonzero solution of Eq. (\(E_{\lambda }\)). Moreover, we have the following results.

Lemma 2.1

Let \(N\ge 5\), \(1\le q<2\) and \(\lambda \in (0,\varLambda _{0})\) where \(\varLambda _{0}\) is the same as in (1.5). Then \(J_{\lambda }\) is coercive and bounded below on \(\mathcal{M}_{\lambda }\).

Proof

If \(u\in \mathcal{M}_{\lambda }\), then by (1.4), (2.1), and the Hölder inequality

$$\begin{aligned} J_{\lambda }(u) &=\frac{1}{2} \Vert u \Vert ^{2}+ \frac{1}{2^{*}} \biggl( \lambda \int _{\varOmega } \vert u \vert ^{q}- \Vert u \Vert ^{2} \biggr)-\frac{\lambda }{q} \int _{\varOmega } \vert u \vert ^{q} \\ &=\frac{2^{*}-2}{22^{*}} \Vert u \Vert ^{2}-\lambda \biggl( \frac{2^{*}-q}{2^{* }q} \biggr) \int _{\varOmega } \vert u \vert ^{q} \end{aligned}$$
(2.2)
$$\begin{aligned} &\geq \frac{2}{N} \Vert u \Vert ^{2}-\lambda \biggl( \frac{2^{*}-q}{2^{*}q} \biggr) \vert \varOmega \vert ^{\frac{2^{*}-q}{2^{*}}}S^{-\frac{q}{2}} \Vert u \Vert ^{q}. \end{aligned}$$
(2.3)

Thus, \(J_{\lambda }\) is coercive and bounded below on \(\mathcal{M} _{\lambda }\). □

Define \(\psi _{\lambda }:H^{2}_{0}(\varOmega )\to \mathbb{R}\), by \(\psi _{\lambda }(u)=\langle J_{\lambda }'( u),u\rangle \), that is,

$$\psi _{\lambda }(u)= \Vert u \Vert ^{2}- \int _{\varOmega } \vert u \vert ^{2^{*}}-\lambda \int _{\varOmega } \vert u \vert ^{q}. $$

Then we see that \(\psi _{\lambda }\in C^{1}(H_{0}^{2}(\varOmega ), \mathbb{R})\), \(\mathcal{M}_{\lambda }=\psi _{\lambda }^{-1}(0)\setminus \{0\}\), and for all \(u\in \mathcal{M}_{\lambda }\),

$$\begin{aligned} \bigl\langle \psi _{\lambda }'(u),u\bigr\rangle &= 2 \Vert u \Vert ^{2}-2^{*} \int _{\varOmega } \vert u \vert ^{2^{*}}-\lambda q \int _{\varOmega } \vert u \vert ^{q} \\ &= (2-q) \Vert u \Vert ^{2}-\bigl(2^{*}-q\bigr) \int _{\varOmega } \vert u \vert ^{2^{*}} \end{aligned}$$
(2.4)
$$\begin{aligned} &= \bigl(2-2^{*}\bigr) \Vert u \Vert ^{2}-\lambda \bigl(q-2^{*} \bigr) \int _{\varOmega } \vert u \vert ^{q}. \end{aligned}$$
(2.5)

We split \(\mathcal{M} _{\lambda }\) into three parts:

$$\begin{aligned}& \mathcal{M}_{\lambda }^{+} =\bigl\{ u\in \mathcal{M}_{\lambda }: \bigl\langle \psi _{\lambda }'(u) ,u\bigr\rangle >0\bigr\} , \\& \mathcal{M}_{\lambda }^{0} =\bigl\{ u\in \mathcal{M}_{\lambda }: \bigl\langle \psi _{\lambda }'(u) ,u\bigr\rangle =0\bigr\} , \\& \mathcal{M}_{\lambda }^{-} =\bigl\{ u\in \mathcal{M}_{\lambda }: \bigl\langle \psi _{\lambda }'(u) ,u\bigr\rangle < 0\bigr\} . \end{aligned}$$

We now derive some basic properties of \(\mathcal{M}_{\lambda }^{+}\), \(\mathcal{M}_{\lambda }^{0}\) and \(\mathcal{M}_{\lambda }^{-}\).

Lemma 2.2

Assume that \(u_{0}\) is a local minimizer for \(J_{\lambda }\) on \(\mathcal{M}_{\lambda }\) and \(u_{0}\notin \mathcal{M}_{\lambda } ^{0}\). Then \(J_{\lambda }'(u_{0})=0\) in \(H^{-2}(\varOmega )\).

Proof

See [5, Theorem 2.3]. □

Moreover, we have the following result.

Lemma 2.3

If \(\lambda \in (0,\varLambda _{0})\), then \(\mathcal{M}_{\lambda }^{0}= \emptyset \).

Proof

Arguing by contradiction, we assume that there exists a \(\lambda \in (0,\varLambda _{0})\) such that \(\mathcal{M}_{\lambda }^{0}\neq \emptyset \). Then, for \(u\in \mathcal{M}_{\lambda }^{0}\) by (1.4) and (2.4), we have

$$ \frac{2-q}{2^{*}-q} \Vert u \Vert ^{2}= \int _{\varOmega } \vert u \vert ^{2^{*}}\le S^{- \frac{2^{*}}{2}} \Vert u \Vert ^{2^{*}} $$

and so

$$\begin{aligned} \Vert u \Vert \ge \biggl(\frac{2-q}{2^{*}-q} \biggr)^{\frac{1}{2^{*}-2}}S^{ \frac{2^{*}}{2(2^{*}-2)}}. \end{aligned}$$

Similarly, using (1.7), (2.5), and the Hölder inequality, we have

$$\begin{aligned} \Vert u \Vert ^{2}=\lambda \frac{2^{*}-q}{2^{*}-2} \int _{\varOmega } \vert u \vert ^{q} \le \lambda \frac{2^{*}-q}{2^{*}-2} \vert \varOmega \vert ^{\frac{2^{*}-q}{2^{*}}}S ^{-\frac{q}{2}} \Vert u \Vert ^{q}, \end{aligned}$$

which implies

$$\begin{aligned} \Vert u \Vert \leq \biggl[\lambda \frac{2^{*}-q}{2^{*}-2} \vert \varOmega \vert ^{ \frac{2^{*}-q}{2^{*}}}S^{-\frac{q}{2}} \biggr]^{\frac{1}{2-q}}. \end{aligned}$$

Hence, we must have

$$\lambda \ge \biggl(\frac{2-q}{2^{*}-q} \biggr)^{\frac{2-q}{2^{*}-2}} \biggl( \frac{2^{*}-2}{2^{*}-q} \biggr) \vert \varOmega \vert ^{- \frac{2^{*}-q}{2^{*}}}S^{\frac{(2-q)N}{8}+\frac{q}{2}}= \varLambda _{0}, $$

which is a contradiction. This completes the proof. □

For each \(u\in H_{0}^{2}(\varOmega )\setminus \{0\}\), let

$$\begin{aligned} \tau _{\mathrm{max}}= \biggl(\frac{(2-q) \Vert u \Vert ^{2} }{(2^{*}-q)\int _{\varOmega } \vert u \vert ^{2^{*}}} \biggr)^{\frac{1}{2^{*}-2}}>0. \end{aligned}$$

Similar to Lemma 2.7 in [14], we can get the following result.

Lemma 2.4

If \(\lambda \in (0,\varLambda _{0})\), then, for each \(u\in H^{2}_{0}( \varOmega )\setminus \{0\}\), the set \(\{\tau u:\tau >0\}\) intersects \(\mathcal{M}_{\lambda }\) exactly twice. More specifically, there exist a unique \(\tau ^{-}=\tau ^{-}(u)>0\) such that \(\tau ^{-}u\in \mathcal{M} _{\lambda }^{-}\) and a unique \(\tau ^{+}=\tau ^{+}(u)>0\) such that \(\tau ^{+}u\in \mathcal{M}_{\lambda }^{+}\). Moreover, \(\tau ^{+}< \tau _{\mathrm{max}}<\tau ^{-}\) and

$$J_{\lambda }\bigl(\tau ^{+}u\bigr)=\inf_{0\leq \tau \leq \tau _{\mathrm{max}}}J _{\lambda }(\tau u),\qquad J_{\lambda }\bigl(\tau ^{-}u\bigr)= \sup_{\tau \geq \tau _{\mathrm{max}}}J_{\lambda }(\tau u). $$

Proof

The proof is similar to that of [14, Lemma 2.7] and is omitted. □

Existence of ground state solutions in the case of \(1\le q<2\)

First, we remark that it follows from Lemma 2.3 that

$$\mathcal{M}_{\lambda }=\mathcal{M}_{\lambda }^{+}\cup \mathcal{M}_{ \lambda }^{-} $$

for all \(\lambda \in (0,\varLambda _{0})\). Furthermore, by Lemma 2.4 it follows that \(\mathcal{M}_{\lambda }^{+}\) and \(\mathcal{M}_{\lambda }^{-}\) are non-empty and by Lemma 2.1 we may define

$$\alpha _{\lambda } =\inf_{u\in \mathcal{M}_{\lambda }}J_{\lambda }(u) ; \qquad \alpha _{\lambda } ^{+}= \inf_{u\in \mathcal{M}_{\lambda }^{+}}J_{\lambda }(u) ;\qquad \alpha _{\lambda }^{-} =\inf_{u\in \mathcal{M}_{\lambda }^{-}}J_{ \lambda }(u). $$

Lemma 3.1

The following facts hold.

  1. (i)

    If \(\lambda \in (0,\varLambda _{0})\), then \(\alpha _{\lambda }\le \alpha _{\lambda } ^{+}<0\).

  2. (ii)

    If \(\lambda \in (0,\frac{q}{2}\varLambda _{0})\), then \(\alpha _{ \lambda }^{-}>c_{0}\) for some \(c_{0}>0\).

In particular, for each \(\lambda \in (0,\frac{q}{2}\varLambda _{0})\), we have \(\alpha _{\lambda } ^{+}=\alpha _{\lambda } \).

Proof

(i) Let \(u\in \mathcal{M}_{\lambda }^{+}\). By (2.4)

$$\frac{2-q}{2^{*}-q} \Vert u \Vert ^{2}> \int _{\varOmega } \vert u \vert ^{2^{*}} $$

and so

$$\begin{aligned} J_{\lambda }(u) & = \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \Vert u \Vert ^{2} + \biggl(\frac{1}{q}- \frac{1}{2^{*}} \biggr) \int _{\varOmega } \vert u \vert ^{2^{*}} \\ & < \biggl[ \biggl(\frac{1}{2}-\frac{1}{q} \biggr)+ \biggl( \frac{1}{q}- \frac{1}{2^{*}} \biggr) \biggl(\frac{2-q}{2^{*}-q} \biggr) \biggr] \Vert u \Vert ^{2} \\ & =-\frac{(2^{*}-2)(2-q)}{22^{*}q} \Vert u \Vert ^{2}< 0. \end{aligned}$$

Therefore, from the definition of \(\alpha _{\lambda }\) and \(\alpha _{\lambda }^{+}\), we can deduce that \(\alpha _{\lambda }\le \alpha _{\lambda }^{+}<0\).

(ii) Let \(u\in \mathcal{M}_{\lambda }^{-}\). By (2.4)

$$\frac{2-q}{2^{*}-q} \Vert u \Vert ^{2}< \int _{\varOmega } \vert u \vert ^{2^{*}} . $$

Moreover, by (1.4) we have

$$\int _{\varOmega } \vert u \vert ^{2^{*}} \le S^{-\frac{2^{*}}{2}} \Vert u \Vert ^{2^{*}}. $$

This implies

$$\begin{aligned} \Vert u \Vert > \biggl(\frac{2-q}{2^{*}-q} \biggr)^{\frac{1}{2^{*}-2}}S^{ \frac{N}{8}} \quad \text{for all } u\in \mathcal{M}_{\lambda }^{-}. \end{aligned}$$
(3.1)

By (2.3) and (3.1), we have

$$\begin{aligned} J_{\lambda }(u)& \ge \Vert u \Vert ^{q} \biggl[ \frac{2}{N} \Vert u \Vert ^{2-q}-\lambda \biggl( \frac{2^{*}-q}{2^{*}q} \biggr) \vert \varOmega \vert ^{\frac{2^{*}-q}{2^{*}}}S^{- \frac{q}{2}} \biggr] \\ & > \biggl(\frac{2-q}{2^{*}-q} \biggr)^{\frac{q}{2^{*}-2}} S^{\frac{qN}{8}} \biggl[ \frac{2}{N} \biggl(\frac{2-q}{2^{*}-q} \biggr)^{ \frac{2-q}{2^{*}-2}} S^{\frac{(2-q)N}{8}}-\lambda \biggl(\frac{2^{*}-q}{ 2^{*}q} \biggr) \vert \varOmega \vert ^{\frac{2^{*}-q}{2^{*}}}S^{-\frac{q}{2}} \biggr] \\ & = \biggl(\frac{q}{2}\varLambda _{0}-\lambda \biggr) \biggl( \frac{2-q}{2^{*}-q} \biggr)^{\frac{q}{2^{*}-2}} \biggl(\frac{2^{*}-q}{2^{*}q} \biggr) \vert \varOmega \vert ^{\frac{2^{*}-q}{2^{*}}}S^{\frac{(N-4)q}{8}}. \end{aligned}$$

Thus, if \(\lambda \in (0,\frac{q}{2}\varLambda _{0})\), then there exists \(c_{0}>0\) such that

$$J_{\lambda }(u)>c_{0}\quad \text{for all }u\in \mathcal{M}_{\lambda }^{-}. $$

Consequently, this completes the proof. □

Remark 3.2

  1. (i)

    If \(\lambda \in (0,\varLambda _{0})\), then, by (1.7), (2.5), and the Hölder inequality, for each \(u\in \mathcal{M}_{\lambda }^{+}\) we have

    $$\begin{aligned} \Vert u \Vert ^{2} &< \lambda \frac{2^{*}-q}{2^{*}-2} \int _{\varOmega } \vert u \vert ^{q} \\ &\le \lambda \frac{2^{*}-q}{2^{*}-2} \vert \varOmega \vert ^{ \frac{2^{*}-q}{2^{*}}}S^{-\frac{q}{2}} \Vert u \Vert ^{q} \end{aligned}$$

    and so

    $$\begin{aligned} \Vert u \Vert < \biggl[\lambda \frac{2^{*}-q}{2^{*}-2} \vert \varOmega \vert ^{ \frac{2^{*}-q}{2^{*}}}S^{-\frac{q}{2}} \biggr]^{\frac{1}{2-q}}\quad \text{for all }u\in \mathcal{M}_{\lambda }^{+}. \end{aligned}$$
    (3.2)
  2. (ii)

    If \(\lambda \in (0,\frac{q}{2}\varLambda _{0})\), then, by Lemma 2.4 and Lemma 3.1(ii), for each \(u\in \mathcal{M}_{ \lambda }^{-}\) we have

    $$\begin{aligned} J_{\lambda }(u)=\sup_{t\geq 0}J_{\lambda }(t u). \end{aligned}$$

We define the Palais–Smale (indicated simply by the prefix “\((\mathrm{PS})\)-”) sequences, \((\mathrm{PS})\)-values, and \((\mathrm{PS})\)-conditions in \(H_{0}^{2}(\varOmega )\) for \(J_{\lambda }\) as follows.

Definition 3.3

  1. (i)

    For \(c\in \mathbb{R}\), a sequence \(\{u_{n}\}\) is a \((\mathrm{PS})_{c}\)-sequence in \(H_{0}^{2}(\varOmega )\) for \(J_{\lambda }\) if \(J_{\lambda }(u_{n})=c+o_{n}(1)\) and \(J^{\prime }_{\lambda }(u_{n})=o _{n}(1)\) strongly in \(H^{-2}(\varOmega )\) as \(n\rightarrow \infty \).

  2. (ii)

    \(c\in \mathbb{R}\) is a \((\mathrm{PS})\)-value in \(H_{0}^{2}(\varOmega )\) for \(J_{\lambda }\) if there exists a \((\mathrm{PS})_{c}\)-sequence in \(H_{0}^{2}( \varOmega )\) for \(J_{\lambda }\).

  3. (iii)

    \(J_{\lambda }\) satisfies the \((\mathrm{PS})_{c}\)-condition in \(H_{0}^{2}(\varOmega )\) if any \((\mathrm{PS})_{c}\)-sequence \(\{u_{n}\}\) in \(H_{0}^{2}(\varOmega )\) for \(J_{\lambda }\) contains a convergent subsequence.

Now, we use the Ekeland variational principle [12] to get the following results.

Proposition 3.4

  1. (i)

    If \(\lambda \in (0,\varLambda _{0})\), then there exists a \((\mathrm{PS})_{\alpha _{\lambda } }\)-sequence \(\{u_{n}\}\subset \mathcal{M} _{\lambda }\) in \(H_{0}^{2}(\varOmega )\) for \(J_{\lambda }\).

  2. (ii)

    If \(\lambda \in (0,\frac{q}{2}\varLambda _{0})\), then there exists a \((\mathrm{PS})_{\alpha _{\lambda }^{-}}\)-sequence \(\{u_{n}\}\subset \mathcal{M}_{\lambda }^{-}\) in \(H_{0}^{2}(\varOmega )\) for \(J_{\lambda }\).

Proof

The proof is similar to that of [14, Proposition 3.3] and is omitted. □

Now, we establish the existence of a local minimum for \(J_{\lambda }\) on \(\mathcal{M}_{\lambda }\).

Theorem 3.5

Let \(N\ge 5\), \(1\le q<2\) and assume that the condition \((\mathcal{H})\) holds. If \(\lambda \in (0,\varLambda _{0})\), then \(J_{\lambda }\) has a minimizer \(u_{\lambda }\) in \(\mathcal{M}_{\lambda }^{+}\) and we have the following results.

  1. (i)

    \(J_{\lambda }(u_{\lambda })=\alpha _{\lambda } =\alpha _{\lambda }^{+}\).

  2. (ii)

    \(u_{\lambda }\) is a nontrivial solution of Eq. (\(E_{\lambda }\)).

  3. (iii)

    \(\|u_{\lambda }\|\to 0\) as \(\lambda \to 0^{+}\).

Proof

By Proposition 3.4(i), there is a minimizing sequence \(\{u_{n}\}\) for \(J_{\lambda }\) on \(\mathcal{M}_{\lambda }\) such that

$$ J_{\lambda }(u_{n})=\alpha _{\lambda } +o_{n}(1) \quad \text{and}\quad J_{\lambda }'(u_{n})=o_{n}(1) \quad \text{in }H^{-2}(\varOmega ). $$
(3.3)

Since \(J_{\lambda }\) is coercive on \(\mathcal{M}_{\lambda }\) (see Lemma 2.1), we see that \(\{u_{n}\}\) is bounded in \(H_{0}^{2}(\varOmega )\). Thus, passing a subsequence if necessary, there exists \(u_{\lambda } \in H_{0}^{2}(\varOmega )\) such that as \(n\to \infty \)

$$ \textstyle\begin{cases} u_{n}\rightharpoonup u_{\lambda }& \text{weakly in }H^{2}_{0}(\varOmega ), \\ u_{n}\to u_{\lambda } &\text{strongly in }L^{q}( \varOmega )\text{ for }1\le q< 2^{*}, \\ u_{n}\to u_{\lambda }&\text{almost everywhere in } \varOmega . \end{cases} $$
(3.4)

It follows that

$$\begin{aligned} \lambda \int _{\varOmega } \vert u_{n} \vert ^{q} \to \lambda \int _{\varOmega } \vert u_{ \lambda } \vert ^{q} \quad \text{as } n\to \infty , \forall 1\le q< 2. \end{aligned}$$
(3.5)

By (3.3), (3.4) and (3.5), it is easy to see that \(u_{\lambda }\) is a weak solution of Eq. (\(E_{\lambda }\)). From \(\{u_{n}\} \subset \mathcal{M}_{\lambda }\), (2.2) and (3.5), we deduce that

$$\begin{aligned} J_{\lambda }(u_{n}) & = \frac{2^{*}-2}{2 2^{*}} \Vert u_{n} \Vert ^{2}-\lambda \biggl(\frac{2^{*}-q}{2^{*}q} \biggr) \int _{\varOmega } \vert u_{n} \vert ^{q} \\ & \geq -\lambda \biggl(\frac{2^{*}-q}{2^{*}q} \biggr) \int _{\varOmega } \vert u _{n} \vert ^{q} \\ &\to -\lambda \biggl(\frac{2^{*}-q}{2^{*}q} \biggr) \int _{\varOmega } \vert u_{ \lambda } \vert ^{q}. \end{aligned}$$

This and \(J_{\lambda }(u_{n})\to \alpha _{\lambda }<0\) (see Lemma 3.1(i)) yield \(\int _{\varOmega }|u_{\lambda }|^{q}>0\), that is, \(u_{\lambda }\not \equiv 0\). We use \(J_{\lambda }(u_{\lambda })=J_{\lambda }(|u_{\lambda }|)\) and \(|u_{\lambda }|\in \mathcal{M}_{\lambda }\). Thus by Lemma 2.2, we may assume that \(u_{\lambda }\) is a nontrivial nonnegative solution of Eq. (\(E_{\lambda }\)).

Now we prove that up to a subsequence, \(u_{n}\to u_{\lambda }\) strongly in \(H_{0}^{2}(\varOmega )\) and \(J_{\lambda }( u_{\lambda })=\alpha _{ \lambda } \). From the fact \(u_{n}\), \(u\in \mathcal{M}_{\lambda }\) and Fatou’s lemma, we have

$$\begin{aligned} \alpha _{\lambda } & \le J_{\lambda }(u_{\lambda })= \frac{2^{*}-2}{22^{*}} \Vert u_{\lambda } \Vert ^{2} -\lambda \biggl( \frac{2^{*}-q}{2^{*}q} \biggr) \int _{\varOmega } \vert u_{\lambda } \vert ^{q} \\ & \le \liminf_{n\to \infty } \biggl[\frac{2^{*}-2}{22^{*}} \Vert u_{n} \Vert ^{2} -\lambda \biggl(\frac{2^{*}-q}{2^{*}q} \biggr) \int _{\varOmega } \vert u _{n} \vert ^{q} \biggr] \\ & = \liminf_{n\to \infty } J_{\lambda }(u_{n}) \\ &=\alpha _{\lambda }, \end{aligned}$$

which implies that \(J_{\lambda }( u_{\lambda })=\alpha _{\lambda }\) and \(\lim_{n\to \infty }\|u_{n}\|^{2}=\|u_{\lambda }\|^{2}\). Standard argument shows that \(u_{n}\to u_{\lambda }\) strongly in \(H^{2}_{0}( \varOmega )\).

Next, we claim \(u_{\lambda }\in \mathcal{M}_{\lambda }^{+}\). Indeed, if \(u_{\lambda }\in \mathcal{M}_{\lambda }^{-}\), by Lemma 2.4, there exist unique \(\tau _{\lambda }^{+}\) and \(\tau _{\lambda }^{-}\) such that \(\tau _{\lambda }^{+}u_{\lambda }\in \mathcal{M}_{\lambda }^{+}\), \(\tau _{\lambda }^{-}u_{\lambda }\in \mathcal{M}_{\lambda }^{-}\) and \(\tau _{\lambda }^{+}<\tau _{\lambda }^{-}=1\). Since

$$\frac{d}{d\tau }J_{\lambda }\bigl(\tau _{\lambda }^{+}u_{\lambda } \bigr)=0\quad \text{and}\quad \frac{d^{2}}{d\tau ^{2}}J_{\lambda }\bigl(\tau _{\lambda } ^{+}u_{\lambda }\bigr)>0, $$

there exists \(\bar{\tau }\in (\tau _{\lambda }^{+},\tau _{\lambda }^{-})\) such that \(J_{\lambda }(\tau _{\lambda }^{+}u_{\lambda })< J_{\lambda }(\bar{ \tau }u_{\lambda })\). By Lemma 2.4 we get

$$J_{\lambda }\bigl(\tau _{\lambda }^{+}u_{\lambda } \bigr)< J_{\lambda }(\bar{ \tau } u_{\lambda })\leq J_{\lambda }\bigl( \tau _{\lambda }^{-}u_{\lambda }\bigr) =J_{\lambda }(u_{\lambda }), $$

which contradicts \(J_{\lambda }(u_{\lambda })=\alpha _{\lambda }\). Consequently, \(u_{\lambda }\in \mathcal{M}_{\lambda }^{+}\).

Finally, by \(u_{\lambda }\in \mathcal{M}_{\lambda }^{+}\) and (3.2), we obtain

$$\Vert u_{\lambda } \Vert < \biggl[\lambda \frac{2^{*}-q}{2^{*}-2} \vert \varOmega \vert ^{ \frac{2^{*}-q}{2^{*}}}S^{-\frac{q}{2}} \biggr]^{\frac{1}{2-q}}\quad \text{for all }u\in \mathcal{M}_{\lambda }^{+}. $$

This implies that \(\|u_{\lambda }\| \to 0\) as \(\lambda \to 0^{+}\), and completes the proof. □

Multiplicity of nontrivial solutions in the case of \(1\le q<2\)

In this section, we will establish the existence of the second nontrivial solution of Eq. (\(E_{\lambda }\)) by proving that \(J_{\lambda }\) attains a local minimum on \(\mathcal{M}_{\lambda }^{-}\).

Lemma 4.1

If \(\{u_{n}\}\subset H_{0}^{2}(\varOmega )\) is a \((\mathrm{PS})_{c}\)-sequence for \(J_{\lambda }\), then \(\{u_{n}\}\) is bounded in \(H_{0}^{2}(\varOmega )\).

Proof

The proof is similar to that of [15, Lemma 4.1] and is omitted. □

We recall that

$$A_{\mu _{i}}:=\inf_{u\in H_{0}^{2}(\varOmega )\setminus \{0\}}\frac{ \int _{\varOmega } ( \vert \Delta u \vert ^{2}-\mu _{i}\frac{u^{2}}{ \vert x-a_{i} \vert ^{4}} )\,dx}{ (\int _{\varOmega } \vert u \vert ^{2^{*}}\,dx )^{\frac{2}{2^{*}}}}. $$

Lemma 4.2

Let \(N\ge 5\), \(1\le q<2\) and assume that \((\mathcal{H})\) holds. If \(\{u_{n}\}\subset H_{0}^{2}(\varOmega )\) is a \((\mathrm{PS})_{c}\)-sequence for \(J_{\lambda }\) with \(c\in (0,\frac{2}{N}A_{\mu _{k}}^{\frac{N}{4}} )\), then there exists a subsequence of \(\{u_{n}\}\) converging weakly to a nonzero solution of Eq. (\(E_{\lambda }\)).

Proof

Let \(\{u_{n}\}\subset H^{2}_{0}(\varOmega )\) be a \((\mathrm{PS})_{c}\)-sequence for \(J_{\lambda }\) with \(c\in (0,\frac{2}{N}A_{\mu _{k}}^{\frac{N}{4}} )\). We know from Lemma 4.1 that \(\{u_{n}\}\) is bounded in \(H_{0}^{2}(\varOmega )\). Then there exists a subsequence of \(\{u_{n}\}\) (still denoted by \(\{u_{n}\}\)) and \(u_{0}\in H_{0}^{2}(\varOmega )\) such that \(u_{n}\rightharpoonup u_{0}\) in \(H^{2}_{0}(\varOmega )\), \(u_{n} \to u_{0}\) almost everywhere in Ω, and \(u_{n}\to u_{0}\) in \(L^{q}( \varOmega )\) for any \(1\le q<2^{*}\) as \(n\to \infty \). It is easy to see that \(J_{\lambda }'(u_{0})=0\) and

$$\begin{aligned} \lambda \int _{\varOmega } \vert u_{n} \vert ^{q}= \lambda \int _{\varOmega } \vert u_{0} \vert ^{q}+o _{n}(1). \end{aligned}$$
(4.1)

Next we verify that \(u_{0}\not \equiv 0\). Arguing by contradiction, we assume \(u_{0}\equiv 0\). By the concentration compactness principle (see [20, 21]) there exists a subsequence, still denoted by \(\{u_{n}\}\), an at most countable set \(\mathcal{J}\), a set of different points \(\{x_{j}\}_{j\in \mathcal{J}}\subset \varOmega \setminus \{a_{1},a _{2},\ldots ,a_{k}\}\), nonnegative real numbers \(\widetilde{\mu _{x_{j}}}\), \(\widetilde{\nu _{x_{j}}}\), \(j\in \mathcal{J}\) and \(\widetilde{\mu _{a_{i}}}\), \(\widetilde{\gamma _{a_{i}}}\), \(\widetilde{\nu _{a_{i}}}\) (\(1\le i\le k\)) such that

$$ \textstyle\begin{cases} \vert \Delta u_{n} \vert ^{2}\rightharpoonup d\widetilde{\mu }\ge \vert \Delta u _{0} \vert ^{2}+\sum_{j\in \mathcal{J}} \widetilde{\mu _{x_{j}}}\delta _{x _{j}}+\sum _{i=1}^{k}\widetilde{\mu _{a_{i}}}\delta _{a_{i}}, \\ \mu _{i} \frac{u_{n}^{2}}{ \vert x-a_{i} \vert ^{4}}\rightharpoonup d \widetilde{\gamma _{a_{i}}}=\mu _{i} \frac{u_{0}^{2}}{ \vert x-a_{i} \vert ^{4}}+ \widetilde{\gamma _{a_{i}}}\delta _{a_{i}}, \\ \vert u_{n} \vert ^{2^{*}}\rightharpoonup d\widetilde{ \nu }= \vert u_{0} \vert ^{2^{*}}+\sum _{j\in \mathcal{J}} \widetilde{\nu _{x_{j}}}\delta _{x_{j}}+\sum_{i=1}^{k} \widetilde{ \nu _{a_{i}}}\delta _{a_{i}}, \end{cases} $$
(4.2)

where \(\delta _{x}\) is the Dirac mass at x. By the Rellich inequalities, we get

$$\begin{aligned} \widetilde{\mu _{a_{i}}}-\mu _{i}\widetilde{\gamma _{a_{i}}}\ge A_{\mu _{i}}\widetilde{\nu _{a_{i}}}^{\frac{2}{2^{*}}}, \quad 1\le i\le k. \end{aligned}$$

Claim 1. We claim that \(\mathcal{J}\) is finite and for any \(j\in \mathcal{J}\), either

$$\widetilde{\nu _{x_{j}}}=0\quad \text{or}\quad \widetilde{\nu _{x_{j}}}\ge A_{0}^{\frac{N}{4}}. $$

In fact, let \(\varepsilon >0\) be small enough such that \(a_{i}\notin B_{2\varepsilon }(x_{j})\) for all \(1\le i\le k\) and \(B_{2\varepsilon }(x_{i})\cap B_{2\varepsilon }(x_{j})=\varnothing \) for \(i\neq j\), \(i,j \in \mathcal{J}\). Let \(\phi _{\varepsilon }^{j}\) be a smooth cut-off function centered at \(x_{j}\) such that \(0\le \phi _{\varepsilon }^{j} \le 1\), \(\phi _{\varepsilon }^{j}=1\) for \(|x-x_{j}|\le \varepsilon \), \(\phi _{\varepsilon }^{j}=0\) for \(|x-x_{j}|\ge 2\varepsilon \), \(|\nabla \phi _{\varepsilon }^{j}|\le \frac{2}{\varepsilon }\) and \(|\Delta \phi _{\varepsilon }^{j}|\le \frac{2}{\varepsilon ^{2}}\). Consider the sequence \(\{\phi _{\varepsilon }^{j}u_{n}\}\); it is obvious that this sequence is bounded in \(H^{2}_{0}(\varOmega )\). Then (4.1) implies

$$\lim_{n\to \infty }\bigl\langle J_{\lambda }'(u_{n}), \phi _{\varepsilon } ^{j}u_{n}\bigr\rangle =0. $$

Moreover, by (4.2) we deduce

$$\begin{aligned} \int _{\varOmega }\sum_{i=1}^{k} \phi _{\varepsilon }^{j} \,d\widetilde{\gamma _{a_{i}}}+ \int _{\varOmega }\phi _{\varepsilon }^{j}\,d\widetilde{\nu }+\lambda \int _{\varOmega } \vert u_{0} \vert ^{q} \phi _{\varepsilon }^{j}\,dx= \lim_{n\to \infty } \int _{\varOmega }\Delta u_{n}\Delta \bigl(u_{n} \phi _{\varepsilon }^{j}\bigr)\,dx. \end{aligned}$$
(4.3)

Then

$$ \textstyle\begin{cases} \lim_{\varepsilon \to 0} \int _{\varOmega }\sum_{i=1}^{k} \phi _{\varepsilon }^{j}\,d\widetilde{\gamma }=\lim_{\varepsilon \to 0} \int _{\varOmega }\sum_{i=1}^{k} \mu _{i} \frac{u_{0}^{2}\phi _{\varepsilon }^{j}}{ \vert x-a_{i} \vert ^{4}}=0, \\ \lim_{\varepsilon \to 0} \int _{\varOmega }\phi _{\varepsilon }^{j}\,d\widetilde{\nu }=\lim_{\varepsilon \to 0} ( \int _{\varOmega } \vert u_{0} \vert ^{2^{*}} \phi _{\varepsilon }^{j}+\widetilde{\nu _{x_{j}}} )= \widetilde{\nu _{x_{j}}}, \\ \lim_{\varepsilon \to 0}\lambda \int _{\varOmega } \vert u_{0} \vert ^{q} \phi _{\varepsilon }^{j}\,dx=0. \end{cases} $$
(4.4)

On the other hand, by (4.2) and the weak convergence we can obtain

$$ \begin{aligned} \lim_{n\to \infty } \int _{\varOmega }\Delta u_{n}\Delta \bigl(u_{n} \phi _{\varepsilon }^{j}\bigr)\,dx = \int _{\varOmega } \phi _{\varepsilon }^{j}\,d\widetilde{\mu } +\lim_{n\to \infty } \int _{\varOmega }\Delta u_{n} \bigl(2\nabla u_{n} \nabla \phi _{\varepsilon }^{j}+u_{n} \Delta \phi _{\varepsilon }^{j} \bigr)\,dx. \end{aligned} $$
(4.5)

Now, by (4.2) it is easy to see that

$$ \begin{aligned} \lim_{\varepsilon \to 0} \int _{\varOmega } \phi _{\varepsilon }^{j}\,d\widetilde{\mu }\ge \widetilde{\mu _{x_{j}}}. \end{aligned} $$
(4.6)

By the Hölder inequality, we get

$$\begin{aligned} 0 \le &\overline{\lim_{n\to \infty }} \biggl\vert \int _{\varOmega }\Delta u_{n} \bigl( \nabla u_{n} \nabla \phi _{\varepsilon }^{j}\bigr)\,dx \biggr\vert \\ \le &\overline{ \lim_{n\to \infty }} \biggl[ \biggl( \int _{\varOmega } \vert \Delta u_{n} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\varOmega } \vert \nabla u_{n} \vert ^{2} \bigl\vert \nabla \phi _{\varepsilon }^{j} \bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggr] \\ \le& C \biggl( \int _{B_{2\varepsilon }(x_{j})} \vert \nabla u_{0} \vert ^{2} \bigl\vert \nabla \phi _{\varepsilon }^{j} \bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \\ \le& C \biggl( \int _{B_{2\varepsilon }(x_{j})} \bigl\vert \nabla \phi _{\varepsilon }^{j} \bigr\vert ^{N}\,dx \biggr)^{\frac{1}{N}} \biggl( \int _{B_{2\varepsilon }(x_{j})} \vert \nabla u_{0} \vert ^{ \frac{2N}{N-2}}\,dx \biggr)^{\frac{N-2}{2N}} \\ \le& C \biggl( \int _{B_{2\varepsilon }(x_{j})} \vert \nabla u_{0} \vert ^{\frac{2N}{N-2}}\,dx \biggr)^{\frac{N-2}{2N}}\to 0 \quad \text{as }\varepsilon \to 0 \end{aligned}$$
(4.7)

and

$$ \begin{aligned}[b] 0 & \le \overline{\lim_{n\to \infty }} \biggl\vert \int _{\varOmega }\Delta u_{n} u_{n} \Delta \phi _{\varepsilon }^{j} \,dx \biggr\vert \\ &\le \overline{ \lim_{n\to \infty }} \biggl[ \biggl( \int _{\varOmega } \vert \Delta u_{n} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\varOmega } \bigl\vert \Delta \phi _{\varepsilon } ^{j} \bigr\vert ^{2} \vert u_{n} \vert ^{2} \,dx \biggr)^{\frac{1}{2}} \biggr] \\ &\le C \biggl( \int _{B_{2\varepsilon }(x_{j})} \bigl\vert \Delta \phi _{\varepsilon }^{j} \bigr\vert ^{2} \vert u_{0} \vert ^{2} \,dx \biggr)^{\frac{1}{2}} \\ &\le C \biggl( \int _{B_{2\varepsilon }(x_{j})} \bigl\vert \Delta \phi _{\varepsilon }^{j} \bigr\vert ^{ \frac{N}{2}}\,dx \biggr)^{\frac{2}{N}} \biggl( \int _{B_{2\varepsilon }(x_{j})} \vert u_{0} \vert ^{\frac{2N}{N-4}}\,dx \biggr)^{\frac{N-4}{2N}} \\ &\le C \biggl( \int _{B_{2\varepsilon }(x_{j})} \vert u_{0} \vert ^{2^{*}}\,dx \biggr)^{ \frac{1}{2^{*}}}\to 0 \quad \text{as }\varepsilon \to 0. \end{aligned} $$
(4.8)

Thus, from (4.3)–(4.8) it follows that

$$\begin{aligned} \widetilde{\mu _{x_{j}}}\le \widetilde{\nu _{x_{j}}}. \end{aligned}$$

By the Sobolev inequality, \(S_{0}\widetilde{\nu _{x_{j}}}^{ \frac{2}{2^{*}}}\le \widetilde{\mu _{x_{j}}}\), hence we deduce that

$$\widetilde{\nu _{x_{j}}}=0\quad \text{or}\quad \widetilde{\nu _{x_{j}}}\ge A_{0}^{\frac{N}{4}}, $$

which implies that \(\mathcal{J}\) is finite. Claim 1 is proved.

Claim 2. We claim that

$$\text{for each}\quad i=1,2,\ldots , k\quad \text{either}\quad \widetilde{\nu _{a_{i}}}=0\quad \text{or}\quad \widetilde{\nu _{a_{i}}}\ge A_{\mu _{i}}^{\frac{N}{4}}. $$

In order to prove claim 2, for each \(i=1,2,\ldots , k\), we consider the possibility of concentration at points \(a_{i}\) \((1\le i\le k)\). For \(\varepsilon >0\) be small enough such that \(x_{j}\notin B_{\varepsilon }(a_{i})\) for all \(j\in \mathcal{J}\) and \(B_{\varepsilon }(a_{i}) \cap B_{\varepsilon }(a_{j})=\varnothing \) for \(i\neq j\) and \(1\le i,j\le k\). Let \(\varphi _{\varepsilon }^{i}\) be a smooth cut-off function centered at \(a_{i}\) such that \(0\le \varphi _{\varepsilon } ^{i}\le 1\), \(\varphi _{\varepsilon }^{i}=1\) for \(|x-a_{i}|\le \varepsilon \), \(\varphi _{\varepsilon }^{i}=0\) for \(|x-a_{i}|\ge 2\varepsilon \), \(|\nabla \varphi _{\varepsilon }^{i}|\le \frac{2}{\varepsilon }\) and \(|\Delta \varphi _{\varepsilon }^{i}|\le \frac{2}{\varepsilon ^{2}}\). Then, by (4.2) and similar arguments to the proof of claim 1, we obtain

$$ \begin{aligned} & \lim_{\varepsilon \to 0}\lim _{n\to \infty } \int _{\varOmega }\Delta u _{n}\Delta \bigl(u_{n} \varphi _{\varepsilon }^{j}\bigr)=\lim_{\varepsilon \to 0} \int _{\varOmega }\varphi _{\varepsilon }^{i}\,d\widetilde{\mu } \ge \lim_{\varepsilon \to 0} \biggl( \int _{\varOmega } \vert \Delta u_{0} \vert ^{2} \varphi _{\varepsilon }^{i}+\widetilde{\mu _{a_{i}}} \biggr)= \widetilde{\mu _{a_{i}}}, \\ & \lim_{\varepsilon \to 0}\lim_{n\to \infty } \int _{\varOmega }\mu _{i}\frac{u_{n}^{2}}{ \vert x-a_{i} \vert ^{4}} \varphi _{\varepsilon }^{i}=\lim_{\varepsilon \to 0} \int _{\varOmega } \varphi _{\varepsilon }^{i}\,d\widetilde{ \gamma } =\lim_{\varepsilon \to 0} \biggl( \int _{\varOmega }\mu _{i}\frac{u_{0}^{2}}{ \vert x-a_{i} \vert ^{4}} \varphi _{\varepsilon }^{i}+\widetilde{\gamma _{a_{i}}} \biggr)= \widetilde{\gamma _{a_{i}}}, \\ & \lim_{\varepsilon \to 0} \lim_{n\to \infty } \int _{\varOmega } \vert u_{n} \vert ^{2^{*}} \varphi _{\varepsilon }^{i}=\lim_{\varepsilon \to 0} \int _{\varOmega }\varphi _{\varepsilon } ^{i}\,d\widetilde{ \nu } =\lim_{\varepsilon \to 0} \biggl( \int _{\varOmega } \vert u _{0} \vert ^{2^{*}} \varphi _{\varepsilon }^{i}+\widetilde{\nu _{a_{i}}} \biggr)= \widetilde{\nu _{a_{i}}}, \\ & \lim_{\varepsilon \to 0} \lim_{n\to \infty } \int _{\varOmega }\mu _{j} \frac{u_{n}^{2}}{ \vert x-a_{j} \vert ^{4}}\varphi _{\varepsilon }^{i}=0\quad \text{for } j\neq i. \end{aligned} $$

Thus we have

$$\begin{aligned} 0=\lim_{\varepsilon \to 0}\lim_{n\to \infty }\bigl\langle J_{\lambda }'(u _{n}), u_{n}\varphi _{\varepsilon }^{i}\bigr\rangle \ge \widetilde{\mu _{a_{i}}}- \mu _{i}\widetilde{\gamma _{a_{i}}}- \widetilde{\nu _{a_{i}}}. \end{aligned}$$

From (4.5) and (4.6) we derive that \(A_{\mu _{i}} \widetilde{\nu _{a_{i}}}^{\frac{2}{2^{*}}}\le \widetilde{\nu _{a_{i}}}\) for all \(1\le i\le k\), and then

$$\text{either}\quad \widetilde{\nu _{a_{i}}}=0 \quad \text{or}\quad \widetilde{\nu _{a_{i}}}\ge A_{\mu _{i}}^{\frac{N}{4}}. $$

Claim 2 is thereby proved.

From the above arguments and (4.1), we conclude that

$$\begin{aligned} c&=\lim_{n\to \infty } \biggl(J_{\lambda }(u_{n})- \frac{1}{2}\bigl\langle J _{\lambda }'(u_{n}), u_{n}\bigr\rangle \biggr) \\ &= \frac{2}{N}\lim_{n\to \infty } \int _{\varOmega } \vert u_{n} \vert ^{2^{*}}+ \biggl(\frac{1}{2}-\frac{1}{q} \biggr)\lambda \int _{\varOmega } \vert u_{0} \vert ^{q} \\ &=\frac{2}{N} \Biggl( \int _{\varOmega } \vert u_{0} \vert ^{2^{*}}+ \sum_{j\in \mathcal{J}}\widetilde{\nu _{x_{j}}}+\sum _{i=1}^{k} \widetilde{\nu _{a_{i}}} \Biggr)+ \biggl(\frac{1}{2}-\frac{1}{q} \biggr)\lambda \int _{\varOmega } \vert u_{0} \vert ^{q} \\ &=\frac{2}{N} \Biggl(\sum_{j\in \mathcal{J}}\widetilde{ \nu _{x_{j}}}+ \sum_{i=1}^{k} \widetilde{\nu _{a_{i}}} \Biggr). \end{aligned}$$

If \(\widetilde{\nu _{a_{i}}}=\widetilde{\nu _{x_{j}}}=0\) for all \(i\in \{1,2,\ldots ,k\}\) and \(j\in \mathcal{J}\), then \(c=0\) which contradicts the assumption that \(c>0\). On the other hand, if there exists an \(i\in \{1,2,\ldots ,k\}\) such that \(\widetilde{\nu _{a_{i}}}\neq 0\) or there exists a \(j\in \mathcal{J}\) with \(\widetilde{\nu _{x_{j}}}\neq 0\), then we infer that

$$c\ge \frac{2}{N}\min \bigl\{ A_{0}^{\frac{N}{4}},A_{\mu _{1}}^{ \frac{N}{4}},A_{\mu _{2}}^{\frac{N}{4}}, \ldots ,A_{\mu _{k}}^{ \frac{N}{4}} \bigr\} =\frac{2}{N}A_{\mu _{k}}^{\frac{N}{4}}, $$

which also contradicts the assumption that \(c<\frac{2}{N}A_{\mu _{k}} ^{\frac{N}{4}}\). Therefore \(u_{0}\) is a nonzero solution of Eq. (\(E_{\lambda }\)). □

Take \(\delta _{0} > 0\) small enough such that \(B_{2\delta _{0}}(a_{k}) \subset \varOmega \). Choose the radial cut-off function \(\eta (x)=\eta (|x|) \in C_{0}^{\infty }(B_{2\delta _{0}}(0))\) such that \(0 \le \eta (x) \le 1\) in \(B_{2\delta _{0}}(0)\) and \(\eta (x)= 1\) in \(B_{\delta _{0}}(0)\). Set \(u_{\varepsilon }(x)= \eta (x-a_{k})y_{\varepsilon }^{\mu _{k}} (x-a _{k})\), where \(y_{\varepsilon }^{\mu _{k}}(x)\) is the same function as in (1.2). The following asymptotic properties hold.

Lemma 4.3

Assume that \(N\ge 5\), \(\mu _{k}\in [0,\bar{\mu })\), \(\delta = \frac{N-4}{2}\) and \(1\leq q<2^{*}\). Then, as \(\varepsilon \to 0\), we have the following estimates:

$$\begin{aligned}& \int _{\varOmega } \biggl( \vert \Delta u_{\varepsilon } \vert ^{2}-\mu _{k}\frac{ \vert u_{ \varepsilon } \vert ^{2}}{ \vert x-a_{k} \vert ^{4}} \biggr)=A_{\mu _{k}}^{\frac{N}{4}}+O \bigl( \varepsilon ^{2(b(\mu _{k})-\delta )}\bigr), \end{aligned}$$
(4.9)
$$\begin{aligned}& \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2^{*}}=A_{\mu _{k}}^{\frac{N}{4}}+O\bigl( \varepsilon ^{2^{*}(b(\mu _{k})-\delta )}\bigr), \end{aligned}$$
(4.10)

and

$$ \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q}= \textstyle\begin{cases} O_{1}(\varepsilon ^{N-q\delta }), &\textit{if } \frac{N}{b( \mu _{k})}< q< 2^{*}, \\ O_{1}(\varepsilon ^{N-q\delta }) \vert \ln \varepsilon \vert ,&\textit{if } q=\frac{N}{b(\mu _{k})}, \\ O_{1}( \varepsilon ^{q(b(\mu _{k})-\delta )}),&\textit{if } 1\le q< \frac{N}{b( \mu _{k})}. \end{cases} $$
(4.11)

Moreover, for all \(N\ge 8\), as \(\varepsilon \to 0\), we have

$$ \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2}= \textstyle\begin{cases} O_{1}(\varepsilon ^{4}), &\textit{if } 0\le \mu _{k}< \mu ^{*}, \\ O_{1}(\varepsilon ^{4} \vert \ln \varepsilon \vert ), &\textit{if } \mu _{k}=\mu ^{*}, \end{cases} $$
(4.12)

where \(\mu ^{*}:=\frac{1}{16}(N^{2}-16)(N^{2}-8N)\).

Proof

See Kang-Xu [19, Lemma 3.2]. □

Lemma 4.4

Let \(N\ge 5\), \(1\le q<2\) and assume that \((\mathcal{H})\) holds. Then, for any \(\lambda >0\), there exists a \(v_{\lambda }\in H_{0}^{2}( \varOmega )\) such that

$$\begin{aligned} \sup_{t\ge 0}J_{\lambda }(t v_{\lambda })< \frac{2}{N}A_{\mu _{k}}^{ \frac{N}{4}}. \end{aligned}$$
(4.13)

In particular, \(\alpha _{\lambda }^{-}< \frac{2}{N}A_{\mu _{k}}^{ \frac{N}{4}}\) for all \(\lambda \in (0,\varLambda _{0})\).

Proof

For \(t\ge 0\), we consider the functions

$$ \begin{aligned} g(t)&:= J_{\lambda }(t u_{\varepsilon }) \\ &=\frac{t^{2}}{2} \Vert u_{ \varepsilon } \Vert ^{2}- \frac{t^{2^{*}}}{2^{*}} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2^{*}}- \lambda \frac{t^{q}}{q} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q} \\ &\le \frac{t^{2}}{2} \int _{\varOmega } \biggl( \vert \Delta u_{\varepsilon } \vert ^{2}- \mu _{k}\frac{u_{\varepsilon }^{2}}{ \vert x-a_{k} \vert ^{4}} \biggr)- \frac{t^{2^{*}}}{2^{*}} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2^{*}} - \lambda \frac{t^{q}}{q} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q} \end{aligned} $$

and

$$\begin{aligned} \bar{g}(t)&:= \frac{t^{2}}{2} \int _{\varOmega } \biggl( \vert \Delta u_{\varepsilon } \vert ^{2}-\mu _{k}\frac{u_{\varepsilon }^{2}}{ \vert x-a_{k} \vert ^{4}} \biggr)-\frac{t ^{2^{*}}}{2^{*}} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2^{*}} \\ &= \frac{t^{2}}{2} \Vert u_{\varepsilon } \Vert _{\mu _{k}}^{2} - \frac{t^{2^{*}}}{2^{*}} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2^{*}}, \end{aligned}$$

where \(\|u_{\varepsilon }\|_{\mu _{k}}^{2}:=\int _{\varOmega } (|\Delta u_{\varepsilon }|^{2}-\mu _{k} \frac{u_{\varepsilon }^{2}}{|x-a_{k}|^{4}} )\).

Using the definitions of g and \(u_{\varepsilon }\), we get

$$g(t)=J_{\lambda }(t u_{\varepsilon })\le \frac{t^{2}}{2} \Vert u_{\varepsilon } \Vert _{\mu _{k}}^{2}, \quad \text{for all } t \ge 0 \text{ and } \lambda >0. $$

Combining this with (4.9), let \(\varepsilon \in (0,1)\), then there exists \(t_{0}\in (0,1)\) not depending on ε such that

$$\begin{aligned} \sup_{0\le t\le t_{0}}g(t) < \frac{2}{N} A_{\mu _{k}}^{\frac{N}{4}}, \quad \text{for all } \lambda >0\text{ and }\varepsilon \in (0,1). \end{aligned}$$
(4.14)

On the other hand, by the fact that

$$\max_{t\ge 0} \biggl(\frac{t^{2}}{2}B_{1}- \frac{t^{2^{*}}}{2^{*}}B_{2} \biggr)=\frac{2}{N}B_{1}^{\frac{N}{4}}B_{2}^{\frac{4-N}{4}}, \quad B_{1}>0, B_{2}>0, $$

and by (4.9) and (4.10), we can get

$$\begin{aligned} \max_{t\ge 0} \bar{g}(t) &=\frac{2}{N} \Vert u_{\varepsilon } \Vert _{\mu _{k}} ^{\frac{N}{4}} \biggl( \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2^{*}} \biggr)^{ \frac{4-N}{4}} \\ &= \frac{2}{N} \bigl(A_{\mu _{k}}^{\frac{N}{4}}+O\bigl( \varepsilon ^{2(b(\mu _{k})-\delta )}\bigr) \bigr)^{\frac{N}{4}} \bigl(A_{\mu _{k}}^{\frac{N}{4}}+O \bigl(\varepsilon ^{2^{*}(b(\mu _{k})-\delta )}\bigr) \bigr)^{ \frac{4-N}{4}} \\ &=\frac{2}{N}A_{\mu _{k}}^{\frac{N}{4}}+O\bigl( \varepsilon ^{2(b(\mu _{k})-\delta )}\bigr). \end{aligned}$$
(4.15)

Hence, for all \(\lambda >0\), \(1\le q<2\), by (4.15) we have

$$ \begin{aligned}[b] \sup_{t\ge t_{0}}g(t)&\le \sup _{t\ge t_{0}} \biggl(\bar{g}(t)-\lambda \frac{t^{q}}{q} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q} \biggr) \\ &\le \frac{2}{N}A_{\mu _{k}}^{\frac{N}{4}}+O\bigl( \varepsilon ^{2(b(\mu _{k})-\delta )}\bigr)-\lambda \frac{t_{0}^{q}}{q} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q}. \end{aligned} $$
(4.16)

Now, we need to distinguish two cases.

  1. Case (i):

    \(1\le q<\frac{N}{b(\mu )}\) and \(q<2\). By (1.3) and (4.11) we have as \(\varepsilon \to 0\)

    $$\int _{\varOmega } \vert u_{\varepsilon } \vert ^{q}=O_{1}\bigl( \varepsilon ^{q(b(\mu )-\delta )}\bigr)>O\bigl( \varepsilon ^{2(b(\mu )-\delta )}\bigr). $$

    Combining this with (4.14) and (4.16), for any \(\lambda >0\), we can choose \(\varepsilon _{\lambda }\) small enough such that

    $$\sup_{t\ge 0}J_{\lambda }(t u_{\varepsilon _{\lambda }})< \frac{2}{N}A _{\mu _{k}}^{\frac{N}{4}}. $$
  2. Case (ii):

    \(\frac{N}{b(\mu )}\le q<2\). By (4.11) we have

    $$ \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q}= \textstyle\begin{cases} O_{1}(\varepsilon ^{N-q\delta }), &\quad \text{if } q>\frac{N}{b( \mu )}, \\ O_{1}(\varepsilon ^{N-q\delta } \vert \ln \varepsilon \vert ), & \quad \text{if } q=\frac{N}{b(\mu )}. \end{cases} $$

    Moreover, it follows from \(b(\mu )>\delta \) and \(q\ge \frac{N}{b( \mu )}\) that

    $$2\bigl(b(\mu )-\delta \bigr)>q\bigl(b(\mu )-\delta \bigr)\ge N-q\delta . $$

    Combining this with (4.14) and (4.16), for any \(\lambda >0\), we can choose \(\varepsilon _{\lambda }\) small enough such that

    $$\sup_{t\ge 0}J_{\lambda }(t u_{\varepsilon _{\lambda }})< \frac{2}{N}A _{\mu _{k}}^{\frac{N}{4}}. $$

From cases (i) and (ii), (4.13) holds by taking \(v_{\lambda }=u _{\varepsilon _{\lambda }}\).

From Lemma 2.4, the definition of \(\alpha _{\lambda }^{-}\) and (4.13), for any \(\lambda \in (0,\varLambda _{0})\), we see that there exists \(t_{\lambda }^{-}>0\) such that \(t_{\lambda }^{-} v_{\lambda } \in \mathcal{M}_{\lambda }^{-}\) and

$$\alpha _{\lambda }^{-}\le J_{\lambda }\bigl(t_{\lambda }^{-} v_{\lambda }\bigr) \le \sup_{t\ge 0}J_{\lambda }(t v_{\lambda })< \frac{2}{N}A_{\mu _{k}} ^{\frac{N}{4}}. $$

The proof is thus completed. □

Now, we establish the existence of a local minimum of \(J_{\lambda }\) on \(\mathcal{M}_{\lambda }^{-}\).

Theorem 4.5

Assume that \(N\ge 5\) and the condition \((\mathcal{H})\) holds. If \(\lambda \in (0,\frac{q}{2}\varLambda _{0})\), then \(J_{\lambda }\) has a minimizer \(U_{\lambda }\) in \(\mathcal{M}_{\lambda }^{-}\) and such that:

  1. (i)

    \(J_{\lambda }(U_{\lambda })=\alpha _{\lambda } ^{-}\).

  2. (ii)

    \(U_{\lambda }\) is a nontrivial solution of Eq. (\(E_{\lambda }\)).

Proof

If \(\lambda \in (0,\frac{q}{2}\varLambda _{0})\), then, by Lemma 3.1(ii), Proposition 3.4(ii) and Lemma 4.4, there exists a \((\mathrm{PS})_{\alpha _{\lambda }^{-}}\)-sequence \(\{u_{n}\}\subset \mathcal{M}_{\lambda }^{-}\) in \(H_{0}^{2}(\varOmega )\) for \(J_{\lambda }\) with \(\alpha _{\lambda }^{-}\in (0,\frac{2}{N}A_{\mu _{k}}^{ \frac{N}{4}} )\). Since \(J_{\lambda }\) is coercive on \(\mathcal{M} _{\lambda }\) (see Lemma 4.1), we see that \(\{u_{n}\}\) is bounded in \(H^{2}_{0}(\varOmega )\). From Lemma 4.2, there exist a subsequence still denoted by \(\{u_{n}\}\) and a nonzero solution \(U_{\lambda } \in H_{0}^{2}(\varOmega )\) of Eq. (\(E_{\lambda }\)) such that \(u_{n}\rightharpoonup U_{\lambda }\) weakly in \(H_{0}^{2}(\varOmega )\).

Now, we first prove that \(U_{\lambda }\in \mathcal{M}_{\lambda }^{-}\). Arguing by contradiction, we assume \(U_{\lambda }\in \mathcal{M}_{ \lambda }^{+}\). Then, by Lemma 2.4, there exists a unique \(t_{\lambda }^{-}\) such that \(t_{\lambda }^{-}U_{\lambda }\in \mathcal{M}_{\lambda }^{-}\). It follows that

$$\alpha _{\lambda }^{-}\le J_{\lambda }\bigl(t_{\lambda }^{-}U_{\lambda } \bigr)< \lim_{n\to \infty }J_{\lambda }\bigl(t_{\lambda }^{-}u_{n} \bigr)\le \lim_{n\to \infty }J_{\lambda }(u_{n})=\alpha _{\lambda }^{-}. $$

This is a contradiction. Consequently, \(U_{\lambda }\in \mathcal{M} _{\lambda }^{-}\).

Next, by the same argument as that in Theorem 3.5, we get \(u_{n}\to U_{\lambda }\) strongly in \(H^{2}_{0}(\varOmega )\) and \(J_{\lambda }( U_{\lambda })=\alpha _{\lambda }^{-}>0\) for all \(\lambda \in (0,\frac{q}{2}\varLambda _{0})\). Since \(J_{\lambda }(U_{ \lambda })=J_{\lambda }(|U_{\lambda }|)\) and \(|U_{\lambda }|\in \mathcal{M}_{\lambda }^{-}\), by Lemma 2.2 we may assume that \(U_{\lambda }\) is a nontrivial nonnegative solution of Eq. (\(E_{\lambda }\)). The proof of this theorem is then completed. □

Proof of Theorem 1.1

The part (i) of Theorem 1.1 immediately follows from Theorem 3.5. When \(0<\lambda <\frac{q}{2}\varLambda _{0}<\varLambda _{0}\), by Theorems 3.5, and 4.5, we see that Eq. (\(E_{\lambda }\)) has at least two nontrivial solutions \(u_{\lambda }\) and \(U_{\lambda }\) such that \(u_{\lambda } \in \mathcal{M}_{\lambda }^{+}\) and \(U_{\lambda }\in \mathcal{M}_{ \lambda }^{-}\). Since \(\mathcal{M}_{\lambda }^{+}\cap \mathcal{M}_{ \lambda }^{-}=\emptyset \), this implies that \(u_{\lambda }\) and \(U_{\lambda }\) are distinct. This completes the proof of Theorem 1.1. □

Existence of solutions in the case of \(2\le q<2^{*}\)

In order to prove Theorem 1.2, we first establish several lemmas.

Lemma 5.1

Let \(N\ge 5\) and assume that \((\mathcal{H})\) holds and one of the following conditions hold:

  1. (i)

    \(\lambda >0\), \(2< q<2^{*}\).

  2. (ii)

    \(0<\lambda <\lambda _{1}\), \(q=2\).

Then the functional \(J_{\lambda }\) satisfies the \((\mathrm{PS})\) condition for all \(c< c^{*}:=\frac{2}{N} A_{\mu _{k}}^{\frac{N}{4}}\).

Proof

The argument is standard and is omitted (e.g. [17]) □

Lemma 5.2

Let \(N\ge 5\) and assume that \((\mathcal{H})\) holds and one of the following conditions holds:

  1. (i)

    \(\lambda >0\), \(\overline{q}< q<2^{*}\), where

    $$\overline{q}=\max \biggl\{ 2, \frac{N}{b(\mu _{k})}, \frac{4(N-2-b( \mu _{k}))}{N-4} \biggr\} . $$
  2. (ii)

    \(N\ge 8\), \(0<\lambda <\lambda _{1}\), \(q=2\), \(0\le \mu _{k} \le \mu ^{*}\).

Then as \(\varepsilon \to 0^{+}\) we have

$$\begin{aligned} \sup_{t\ge 0}J_{\lambda }(t u_{\varepsilon })< c^{*}= \frac{2}{N}A_{\mu _{k}}^{\frac{N}{4}}, \end{aligned}$$
(5.1)

where \(\lambda _{1}\) is the same as in (1.6) and \(u_{\varepsilon }\) is the same function as in Lemma 4.3.

Proof

For \(t\ge 0\), we define the functions \(g(t):=J_{\lambda }(t u_{\varepsilon })\) and

$$\begin{aligned} \bar{g}(t):= &\frac{t^{2}}{2} \int _{\varOmega } \biggl( \vert \Delta u_{\varepsilon } \vert ^{2}-\mu _{k}\frac{u_{\varepsilon }^{2}}{ \vert x-a_{k} \vert ^{4}} \biggr)-\frac{t ^{2^{*}}}{2^{*}} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2^{*}}. \end{aligned}$$

(i) Since \(\lambda >0\), \(2< q<2^{*}\), a direct calculation shows that \(\sup_{t\ge 0}g(t)\) can be obtained at finite \(t_{\varepsilon }>0\) such that

$$0=g'(t_{\varepsilon })=t_{\varepsilon } \biggl( \Vert u_{\varepsilon } \Vert ^{2}-t _{\varepsilon }^{2^{*}-2} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2^{*}}- \lambda t_{\varepsilon }^{q-2} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q} \biggr). $$

Furthermore, \(t_{\varepsilon }\in [C_{1},C_{2}]\), where \(C_{1}\) and \(C_{2}\) are positive constants independent of ε.

From the definitions of g, and (4.15), it follows that

$$\begin{aligned} g(t_{\varepsilon })\le \bar{g}(t_{\varepsilon })-\frac{\lambda }{q} t _{\varepsilon }^{q} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q}\le c^{*}+O\bigl( \varepsilon ^{2(b(\mu _{k})-\delta )}\bigr)-C \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q}. \end{aligned}$$
(5.2)

If \(\overline{q}< q<2^{*}\), by (4.11) we have

$$\begin{aligned} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{q}=O_{1}\bigl(\varepsilon ^{N-q\delta }\bigr). \end{aligned}$$
(5.3)

Since \(2(b(\mu _{k})-\delta ))>N-q\delta \), from (5.2) and (5.3) it follows that

$$\sup_{t\ge 0}J_{\lambda }(t u_{\varepsilon })=g(t_{\varepsilon })< c ^{*}. $$

(ii) Suppose that \(N\ge 8\), \(0<\lambda <\lambda _{1}\), \(q=2\), \(0\le \mu _{k}\le \mu ^{*}\). A direct calculation shows that

$$0\le \mu _{k}\le \mu ^{*}\quad \Longleftrightarrow\quad b(\mu _{k})-\delta >2, \qquad \mu _{k}=\mu ^{*} \quad \Longleftrightarrow\quad b(\mu _{k})-\delta =2. $$

Using a similar argument to (i), we can deduce that \(\sup_{t\ge 0}g(t)< c^{*}\) is attained at finite \(t_{\varepsilon }>0\). Moreover,

$$\begin{aligned} g(t_{\varepsilon })\le \bar{g}(t_{\varepsilon })-\lambda \frac{ t_{ \varepsilon }^{2}}{2} \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2} \le c^{*}+O\bigl( \varepsilon ^{2(b(\mu _{k})-\delta )}\bigr)-C \int _{\varOmega } \vert u_{\varepsilon } \vert ^{2}. \end{aligned}$$
(5.4)

Then by (4.12) and (5.4) it follows that (5.1) holds as \(\varepsilon \to 0^{+}\). The proof is thus completed. □

Proof of Theorem 1.2

According to Lemmas 5.1 and 5.2 and applying the mountain-pass theorem [2, 4], Theorem 1.2 can be concluded to. □

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Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which improved the original manuscript.

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T.S. Hsu was supported by the Ministry of Science and Technology, Taiwan (Grant No. 107-2115-M-182-002-).

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Correspondence to Tsing-San Hsu.

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MSC

  • 47G20
  • 35J50
  • 36B65

Keywords

  • Multiplicity
  • Biharmonic equations
  • Rellich-type potential
  • Critical Sobolev exponent