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Existence of multiple positive solutions for nonhomogeneous fractional Laplace problems with critical growth

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Abstract

We prove the existence of multiple positive solutions of fractional Laplace problems with critical growth by using the method of monotonic iteration and variational methods.

Introduction

Considerable attention has been devoted to fractional and non-local operators of elliptic type in recent years, both for their interesting theoretical structure and in view of concrete applications, like flame propagation, chemical reactions of liquids, population dynamics, geophysical fluid dynamics, and American options; see [3, 7, 19, 20] and the references therein.

In this paper we consider the following critical problem:

$$ (P)_{\gamma } \quad \textstyle\begin{cases} (-\Delta )^{s} u=\lambda u+ \vert u \vert ^{p-2}u+\gamma g(x) & \text{in } \varOmega , \\ u=0 & \text{in } \mathbb{R}^{N}\setminus \varOmega , \end{cases} $$
(1.1)

where \(s\in (0,1)\) is fixed and \((-\Delta )^{s}\) is the fractional Laplace operator, \(\varOmega \subset \mathbb{R}^{N}\) (\(N>2s\)) is a smooth bounded domain, \(p=2^{*}_{s}:=\frac{2N}{N-2s}\), \(g\in C_{0}(\varOmega )\), \(g(x)\ge 0\) a.e. in Ω and \(g(x)\not \equiv 0\) in Ω, \(\lambda \ge 0\), \(\gamma >0\) are some given constants.

We are interested in the existence of positive solutions of \((P)_{\gamma }\) since it exhibits many interesting existence phenomena which are related to some lack of compactness of the corresponding energy functional (see (1.2)). It is worth noting here that the problem \((P)_{\gamma }\), with \(\lambda =0\), \(\gamma =0\), has no positive solution whenever Ω is a star-shaped domain; see [6, 11]. This fact motivates the perturbation terms λu and \(\gamma g(x)\), in our work. Servadei and Valdinoci [14, 15], and Tan [17] studied problem \((P)_{\gamma }\) with \(\gamma =0\) and obtained Brezis–Nirenberg type results. An interesting problem is whether the existence phenomena still remain true if we give \((P)_{\gamma }\) with \(\gamma =0\) a lower order homogeneous perturbation in the sense \(\lim_{u\to 0}\frac{f(x,u)}{u ^{p-1}}=0\) and \(f(x,0)=0\). The existence results have been obtained in [14, 15] for the fractional Laplace operator, and [8] for the fractional p-Laplace operator. We consider here the nonhomogeneous perturbation case. Note that problem \((P)_{\gamma }\) in the local case \(s=1\) has been investigated in [4, 18].

The fractional Laplace operator \((-\Delta )^{s}\) (up to normalization factors) may be defined as

$$ -(-\Delta )^{s}u(x)= \int _{\mathbb{R}^{N}} \bigl(u(x+y)+u(x-y)-2u(x) \bigr)K(y)\,dy, \quad x \in \mathbb{R}^{N}, $$

where \(K(x)=|x|^{-(N+2s)}\), \(x\in \mathbb{R}^{N}\). We will denote by \(H^{s}(\mathbb{R}^{N})\) the usual fractional Sobolev space endowed with the so-called Gagliardo norm,

$$ \Vert u \Vert _{H^{s}(\mathbb{R}^{N})}= \Vert u \Vert _{L^{2}(\mathbb{R}^{N})}+ \biggl( \int _{\mathbb{R}^{2N}} \bigl\vert u(x)-u(y) \bigr\vert ^{2}K(x-y)\,dx\,dy \biggr)^{1/2}, $$

while \(X_{0}\) is the function space defined as

$$ X_{0}= \bigl\{ u\in H^{s}\bigl(\mathbb{R}^{N} \bigr): u=0\text{ a.e. in } \mathbb{R}^{N}\setminus \varOmega \bigr\} . $$

We refer to [9, 12, 13] for a general definition of \(X_{0}\) and its properties. The embedding \(X_{0}\hookrightarrow L^{q}( \varOmega )\) is continuous for any \(q\in [1,2^{*}_{s}]\) and compact for any \(q\in [1,2^{*}_{s})\). The space \(X_{0}\) is endowed with the norm defined as

$$ \Vert u \Vert _{X_{0}}= \biggl( \int _{\mathbb{R}^{2N}} \bigl\vert u(x)-u(y) \bigr\vert ^{2}K(x-y)\,dx\,dy \biggr) ^{1/2}. $$

By Lemma 5.1 in [12] we have \(C^{2}_{0}(\varOmega )\subset X_{0}\). Thus \(X_{0}\) is non-empty. Note that \((X_{0}, \|\cdot \|_{X_{0}})\) is a Hilbert space with scalar product

$$ (u,v)_{X_{0}}= \int _{\mathbb{R}^{2N}}\bigl(u(x)-u(y)\bigr) \bigl(v(x)-v(y)\bigr)\,dx \,dy. $$

We say that \(u\in X_{0}\) is a weak solution of (1.1) if the identity

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(u(x)-u(y)\bigr) \bigl(\varphi (x)-\varphi (y) \bigr)K(x-y)\,dx\,dy\\& \quad = \lambda \int _{\varOmega }u\varphi \,dx+ \int _{\varOmega } \vert u \vert ^{p-2}u\varphi \,dx+ \gamma \int _{\varOmega }g\varphi \,dx \end{aligned}$$

holds for all \(\varphi \in X_{0}\).

We consider the energy functional associated with (1.1)

$$\begin{aligned} I(u) =&\frac{1}{2} \int _{\mathbb{R}^{2N}} \bigl\vert u(x)-u(y) \bigr\vert ^{2}K(x-y)\,dx\,dy- \frac{1}{2}\lambda \int _{\varOmega }u^{2}\,dx \\ &{}-\frac{1}{p} \int _{\varOmega } \vert u \vert ^{p}\,dx- \gamma \int gu\,dx. \end{aligned}$$
(1.2)

The critical points of the functional I correspond to weak solutions of (1.1).

Let \(\lambda _{1}\) be the first eigenvalue of \((-\Delta )^{s}\) on \(X_{0}\). Our main results are as follows.

Theorem 1.1

For \(\lambda \in [0,\lambda _{1})\) there exists a positive constant \(\gamma ^{*}\) such that \((P)_{\gamma }\) admits a positive minimal solution for all \(\gamma \in (0,\gamma ^{*}]\) and admits no positive solution for \(\gamma >\gamma ^{*}\).

We prove Theorem 1.1 by the method of monotonic iteration, also known as the super and subsolution method, which is a basic tool in nonlinear partial differential equations. In this paper, we discuss a fractional Laplace operator version of this method compared with second order linear or quasilinear elliptic operator. With respect to the classical case of the Laplacian, here some estimates are more delicate, due to the non-local nature of the operator \((-\Delta )^{s}\).

Theorem 1.2

For \(\lambda \in [0,\lambda _{1})\), \(\gamma \in (0,\gamma ^{*})\), where \(\gamma ^{*}\) is the one in Theorem 1.1, problem \((P)_{\gamma }\) admits at least two positive solutions.

In order to prove Theorem 1.2, we adapt the variational approach used in [1] to the non-local framework (see also [15]).

This paper is organized as follows. In Sect. 2 we prove the existence of the first solution of \((P)_{\gamma }\) by the method of monotonic iteration. In Sect. 3 we prove the existence of the second solution of \((P)_{\gamma }\) by variational methods. We denote by \(|\cdot |_{p}\) the \(L^{p}(\varOmega )\)-norm for any \(p>1\), respectively.

Existence of the first positive solution

In this section we prove existence of the first solution of \((P)_{\gamma }\) by the method of monotonic iteration.

Definition 1

We say that \(\overline{u}\in X_{0}\) is a weak supersolution of problem \((P)_{\gamma }\) if

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(\overline{u}(x)-\overline{u}(y)\bigr) \bigl( \varphi (x)- \varphi (y)\bigr)K(x-y)\,dx\,dy \\ & \quad \ge \lambda \int _{\varOmega }\overline{u}\varphi \,dx+ \int _{\varOmega } \vert \overline{u} \vert ^{p-2} \overline{u}\varphi \,dx+\gamma \int _{\varOmega }g\varphi \,dx \end{aligned}$$

for any \(\varphi \in X_{0}\), \(\varphi \ge 0\) a.e. in Ω.

Definition 2

We say that \(\underline{u}\in X_{0}\) is a weak subsolution of problem \((P)_{\gamma }\) if

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(\underline{u}(x)-\underline{u}(y)\bigr) \bigl(\varphi (x)- \varphi (y)\bigr) K(x-y)\,dx\,dy\\ & \quad \le \lambda \int _{\varOmega }\underline{u}\varphi \,dx+ \int _{\varOmega } \vert \underline{u} \vert ^{p-2} \underline{u}\varphi \,dx+\gamma \int _{\varOmega }g\varphi \,dx \end{aligned}$$

for any \(\varphi \in X_{0}\), \(\varphi \ge 0\) a.e. in Ω.

Let \(\lambda _{1}\) be the first eigenvalue of \((-\Delta )^{s}\) on \(X_{0}\) with \(\phi _{1}\ge 0\) the corresponding normalized eigenfunction; see Proposition 9 in [13]. We show \(\phi _{1}>0\) in Ω. By Proposition 4 in [14], \(\phi _{1}\in L^{\infty }(\varOmega )\). Furthermore, by Proposition 1.1 in [10], \(\phi _{1}\in C^{s}( \mathbb{R}^{N})\). Assume by contradiction that there exists \(x_{0}\in \varOmega \) such that \(\phi _{1}(x_{0})=0\). It follows from the definition of the fractional Laplace \((-\Delta )^{s}\) that

$$ 0>- \int _{\mathbb{R}^{N}} \bigl(\phi _{1}(x_{0}+y)+ \phi _{1}(x_{0}-y)-2\phi _{1}(x_{0}) \bigr) K(y)\,dy=\lambda _{1}\phi _{1}(x_{0})=0, $$

we get a contradiction. Thus, \(\phi _{1}>0\) in Ω.

Lemma 2.1

For \(\lambda \in [0,\lambda _{1})\) there exists a constant \(\widehat{\gamma }>0\) such that \((P)_{\gamma }\) has no positive solution for \(\gamma >\widehat{\gamma }\).

Proof

Taking \(C_{1}=\min_{t\ge 0} [t^{p-1}-(\lambda _{1}-\lambda )t ]\) we get

$$ t^{p-1}\ge [\lambda _{1}- \lambda ]t+C_{1}, \quad \forall t\ge 0. $$
(2.1)

Multiplying (1.1) by \(\phi _{1}\) and integrating on Ω we get

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(u(x)-u(y)\bigr) \bigl(\phi _{1}(x)-\phi _{1}(y)\bigr)K(x-y)\,dx\,dy\\ & \quad = \lambda \int _{\varOmega }u\phi _{1} \,dx+ \int _{\varOmega }u^{p-1}\phi _{1}\,dx+ \gamma \int _{\varOmega }g\phi _{1}\,dx. \end{aligned}$$

Consequently,

$$ \lambda _{1} \int _{\varOmega }u\phi _{1} \,dx=\lambda \int _{\varOmega }u\phi _{1}\,dx+ \int _{\varOmega }u^{p-1}\phi _{1}\,dx+\gamma \int _{\varOmega }g\phi _{1}\,dx. $$

Hence from (2.1) we have

$$ \gamma \le \widehat{\gamma }:=\frac{-C_{1}\int _{\varOmega }\phi _{1}\,dx}{ \int _{\varOmega }g\phi _{1}\,dx}. $$

 □

Lemma 2.2

Let \(u_{1}, u_{2}\in X_{0}\) be supersolutions of (\(P_{\gamma }\)). Then \(u_{1}\wedge u_{2}:=\min \{u_{1}, u_{2}\}\) is a supersolution of (\(P _{\gamma }\)). Similarly, if \(v_{1}, v_{2}\in X_{0}\) are subsolutions of (\(P_{\gamma }\)), then so is \(v_{1}\vee v_{2}:=\max \{v_{1}, v_{2}\}\).

Proof

By density results for \(X_{0}\), there exists a sequence \(\{w_{n}\} \subset C^{\infty }(\varOmega )\) such that \(w_{n}\to w:=u_{1}-u_{2}\) in \(X_{0}\). It follows that \(w_{n}(x)\to w(x)\) for a.e. \(x\in \varOmega \).

Let \(\eta \in C^{\infty }(\mathbb{R})\) be a nondecreasing function such that (i) \(0\le \eta (t)\le 1\); (ii) \(\eta (t)=0\) for \(t\le 0\), \(\eta (t)=1\) for \(t\ge 1\). Set \(\eta _{n}(t)=\eta (nt)\). Then \(\eta _{n}(t)=0\) for \(t\le 0\), \(\eta _{n}(t)=1 \) for \(t \ge \frac{1}{n}\).

Now for any nonnegative function \(\varphi \in C^{\infty }_{0}(\varOmega )\) we define

$$ \psi _{1,n}=(1-\eta _{n}\circ w_{n})\varphi , \qquad \psi _{2,n}=(\eta _{n}\circ w_{n}) \varphi , $$

where \(\eta _{n}\circ w_{n}\) denotes the composition of \(w_{n}\) and \(g_{n}\). Of course, \(\psi _{1,n},\psi _{2,n}\ge 0\) and \(\varphi =\psi _{1,n}+\psi _{2,n}\). Since \(u_{1}\), \(u_{2}\) are supersolutions of \((P)_{\gamma }\), we have

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(u_{i}(x)-u_{i}(y) \bigr)\bigl[\psi _{i,n}(x)-\psi _{i,n}(y)\bigr]K(x-y)\,dx \,dy \\ & \quad \ge \lambda \int _{\varOmega }u_{i}\psi _{i,n}\,dx+ \int _{\varOmega } \vert u_{i} \vert ^{p-2}u _{i}\psi _{i,n}\,dx+\gamma \int _{\varOmega }g\psi _{i,n}\,dx, \end{aligned}$$

for \(i=1,2\). It follows that

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(u_{1}(x)-u_{1}(y) \bigr) \bigl\{ \bigl[1-\eta _{n}\bigl(w_{n}(x)\bigr) \bigr] \varphi (x)-\bigl[1-\eta _{n}\bigl(w_{n}(y)\bigr) \bigr]\varphi (y) \bigr\} K(x-y)\,dx\,dy \\ & \quad \ge \lambda \int _{\varOmega }u_{1}(x)\bigl[1-\eta _{n} \bigl(w_{n}(x)\bigr)\bigr]\varphi (x)\,dx+ \int _{\varOmega } \vert u_{1} \vert ^{p-2}u_{1}\bigl[1-\eta _{n} \bigl(w_{n}(x)\bigr)\bigr]\varphi (x)\,dx \\ & \qquad {}+\gamma \int _{\varOmega }g\bigl[1-\eta _{n} \bigl(w_{n}(x)\bigr)\bigr]\varphi (x)\,dx \end{aligned}$$
(2.2)

and

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(u_{2}(x)-u_{2}(y) \bigr) \bigl\{ \eta _{n}\bigl(w_{n}(x)\bigr) \varphi (x)- \eta _{n}\bigl(w_{n}(y)\bigr)\varphi (y) \bigr\} K(x-y) \,dx\,dy \\& \quad \ge \lambda \int _{\varOmega }u_{2}(x)\eta _{n} \bigl(w_{n}(x)\bigr)\varphi (x)\,dx+ \int _{\varOmega } \vert u_{2} \vert ^{p-2}u_{2}\eta _{n}\bigl(w_{n}(x) \bigr)\varphi (x)\,dx \\& \qquad {}+ \gamma \int _{\varOmega }g\eta _{n}\bigl(w_{n}(x) \bigr)\varphi (x)\,dx. \end{aligned}$$
(2.3)

For a.e. \(x\in \varOmega _{1}:=\{x\in \varOmega : u_{1}(x)>u_{2}(x)\}\), \(w(x)>0\) and hence \(\eta _{n}(w_{n}(x))\to 1\) for a.e. \(x\in \varOmega _{1}\). Similarly, \(\eta _{n}(w_{n}(x))\to 0\) for a.e. \(x\in \varOmega _{2}:= \{x\in \varOmega : u_{1}(x)< u_{2}(x)\}\). Adding (2.2) and (2.3), we have

$$\begin{aligned}& \int _{\mathbb{R}^{2N}} \bigl[u_{2}(x)-u_{2}(y)- \bigl(u_{1}(x)-u_{1}(y)\bigr) \bigr] \bigl\{ \eta _{n}\bigl(w_{n}(x)\bigr)\varphi (x)-\eta _{n}\bigl(w_{n}(y)\bigr)\varphi (y) \bigr\} K(x-y)\,dx \,dy \\& \qquad {} + \int _{\mathbb{R}^{2N}}\bigl(u_{1}(x)-u_{1}(y) \bigr) \bigl(\varphi (x)-\varphi (y)\bigr)K(x-y)\,dx\,dy \\& \quad \ge \lambda \int _{\varOmega }u_{1}(x)\varphi (x) \,dx+\lambda \int _{\varOmega }\bigl[u _{2}(x)-u_{1}(x) \bigr]\eta _{n}\bigl(w_{n}(x)\bigr)\varphi (x)\,dx+ \int _{\varOmega } \vert u_{1} \vert ^{p-2}u _{1}\varphi \,dx \\& \qquad {}+ \int _{\varOmega }\bigl[ \vert u_{2} \vert ^{p-2}u_{2}- \vert u_{1} \vert ^{p-2}u_{1}\bigr]\eta _{n} \bigl(w_{n}(x)\bigr) \varphi (x)\,dx +\gamma \int _{\varOmega }g\varphi (x)\,dx. \end{aligned}$$
(2.4)

Define

$$\begin{aligned}& A_{1}:=\bigl\{ (x,y)\in \mathbb{R}^{2N}: w(x)>0, w(y)>0 \bigr\} ,\qquad A_{2}:=\bigl\{ (x,y) \in \mathbb{R}^{2N}: w(x)>0, w(y)\le 0\bigr\} , \\& A_{3}:=\bigl\{ (x,y)\in \mathbb{R}^{2N}: w(x)\le 0, w(y)>0\bigr\} ,\qquad A_{4}:=\bigl\{ (x,y) \in \mathbb{R}^{2N}: w(x)\le 0, w(y)\le 0\bigr\} . \end{aligned}$$

By the dominated convergence theorem, we find, as \(n\to \infty \),

Similarly, as \(n\to \infty \),

and

$$\begin{aligned}& \int _{\varOmega } \vert u_{1} \vert ^{p-2}u_{1}\varphi \,dx+ \int _{\varOmega }\bigl[ \vert u_{2} \vert ^{p-2}u _{2}- \vert u_{1} \vert ^{p-2}u_{1}\bigr]\eta _{n} \bigl(w_{n}(x)\bigr)\varphi (x)\,dx\\& \quad \to \int _{ \varOmega } \vert u_{1}\wedge u_{2} \vert ^{p-2}(u_{1}\wedge u_{2})\varphi \,dx. \end{aligned}$$

Thus, by (2.4), we obtain

$$\begin{aligned}& \int _{\mathbb{R}^{2N}} \bigl[(u_{1}\wedge u_{2}) (x)-(u_{1}\wedge u _{2}) (y) \bigr]\bigl(\varphi (x)- \varphi (y)\bigr)K(x-y)\,dx\,dy \\& \quad \ge \lambda \int _{\varOmega }(u_{1}\wedge u_{2})\varphi \,dx+ \int _{ \varOmega } \vert u_{1}\wedge u_{2} \vert ^{p-2}(u_{1}\wedge u_{2})\varphi \,dx+ \gamma \int _{\varOmega }g\varphi \,dx \end{aligned}$$

for any \(\varphi \in C^{\infty }_{0}(\varOmega )\) with \(\varphi \ge 0\). Since \(C^{\infty }_{0}(\varOmega )\) is dense in \(X_{0}\), for any \(\varphi \in X_{0}\) with \(\varphi \ge 0\), we can find \(\varphi _{n} \in C^{\infty }_{0}\) such that \(\varphi _{n}\to \varphi \) in the \(X_{0}\) norm. This completes the proof. □

Remark 2.3

Lemma 2.2 is valid for the following second order quasilinear elliptic operator in divergence form:

$$ \sum^{N}_{i=1}D_{i} \bigl(A_{i}\bigl(x,u(x),Du(x)\bigr)\bigr), $$

where \(A_{i}\) (\(i=1,\ldots , N\)) satisfies some conditions; see [5] for more details.

Lemma 2.4

For any \(\lambda \in [0,\lambda _{1})\) problem \((P)_{\gamma }\) admits at least one positive solutions which is a minimum of all solutions if γ is small enough.

Proof

Set

$$ \varepsilon =\frac{1}{2}\frac{(\lambda _{1}-\lambda )^{1/(p-2)}}{ \max _{x\in \varOmega }\phi _{1}(x)} $$

and

$$ \rho =\frac{\inf _{x\in \operatorname{supp} g} \{ [ \lambda _{1}-\lambda ]\varepsilon \phi _{1}-(\varepsilon \phi _{1})^{p-1} \} }{\sup _{x\in \varOmega }g(x)}, $$

where \(\text{supp}~g\) denotes the closure of \(\{x\in \varOmega |g(x) \ne 0\}\). It is easy to verify that \(\overline{u}=\varepsilon \phi _{1}\) is a supersolution of \((P)_{\gamma }\) if \(\gamma \le \rho \) and \(\underline{u}=0\) is a subsolution of \((P)_{\gamma }\) for all \(\gamma \ge 0\).

Now let \(u_{0}=\underline{u}\), and then given \(u_{n}\) inductively define \(u_{n+1}\) to be the unique weak solution of linear boundary value problem

$$ \textstyle\begin{cases} (-\Delta )^{s} u_{n+1}=\lambda u_{n}+ \vert u_{n} \vert ^{p-2}u_{n}+\gamma g & \text{in } \varOmega , \\ u_{n+1}=0 & \text{in } \mathbb{R}^{N}\setminus \varOmega . \end{cases} $$
(2.5)

Similarly let \(w_{0}=\overline{u}\), and then given \(w_{n}\) inductively define \(w_{n+1}\) to be the unique weak solution of linear boundary value problem

$$ \textstyle\begin{cases} (-\Delta )^{s} w_{n+1} =\lambda w_{n}+ \vert w_{n} \vert ^{p-2}w_{n}+\gamma g & \text{in } \varOmega , \\ w_{n+1}=0 & \text{in } \mathbb{R}^{N}\setminus \varOmega . \end{cases} $$
(2.6)

Claim 1. \(\underline{u}=u_{0}\le u_{1}\le w_{1}\le w_{0}=\overline{u}\).

From (2.5) we deduce

$$ \int _{\mathbb{R}^{2N}}\bigl(u_{1}(x)-u_{1}(y) \bigr) \bigl(\varphi (x)-\varphi (y)\bigr)K(x-y)\,dx\,dy= \gamma \int _{\varOmega }g\varphi \,dx, \quad \forall \varphi \in X_{0}. $$
(2.7)

Similarly from (2.6) we have

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(w_{1}(x)-w_{1}(y) \bigr) \bigl(\varphi (x)-\varphi (y)\bigr)K(x-y)\,dx\,dy \\& \quad = \lambda \int _{\varOmega }\overline{u}\varphi \,dx+ \int _{\varOmega } \overline{u}^{p-1}\varphi \,dx+\gamma \int _{\varOmega }g\varphi \,dx,\quad \forall \varphi \in X_{0}. \end{aligned}$$
(2.8)

Subtract (2.8) from (2.7) and set \(\varphi =(u_{1}-w_{1})^{+}\). We obtain

$$ \int _{\mathbb{R}^{2N}}\bigl[\psi _{1}(x)-\psi _{1}(y)\bigr] \bigl[\psi _{1}^{+}(x)-\psi _{1}^{+}(y)\bigr]K(x-y)\,dx\,dy\le 0, $$
(2.9)

where \(\psi _{1}(x)=u_{1}(x)-w_{1}(x)\), for all \(x\in \mathbb{R}^{N}\). It is easy to see that

$$ \bigl[\psi _{1}(x)-\psi _{1}(y)\bigr] \bigl[\psi _{1}^{+}(x)-\psi _{1}^{+}(y) \bigr]\ge \bigl\vert \psi ^{+}(x)-\psi ^{+}(y) \bigr\vert ^{2}, \quad \forall x,y\in \mathbb{R}^{N}. $$

So, by (2.9),

$$ \psi _{1}^{+}(x)-\psi _{1}^{+}(y)=0, \quad \forall x,y\in \mathbb{R}^{N}. $$

Then, \(\psi _{1}^{+}(x)=0\) for all \(x\in \mathbb{R}^{N}\) since \(\psi (x)=0\) for any \(x\in \mathbb{R}^{N}\setminus \varOmega \). So \(\psi _{1}\le 0\) and \(u_{1}\le w_{1}\) a.e. in Ω.

Similarly, by the definition of supersolution and subsolution, (2.5) and (2.6) we can prove \(u_{0}\le u_{1}\) and \(w_{1}\le w_{0}\).

Claim 2. \(u_{n}\le u_{n+1}\le w_{n+1}\le w_{n}\) a.e. in Ω, \(\forall n=0,1,2,\ldots \) .

Claim 2 obviously holds for \(n=0\). Assume for induction that

$$ u_{n-1}\le u_{n}\le w_{n}\le w_{n-1} \quad \text{a.e. in }\varOmega . $$

From (2.5) and (2.6) we have

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(u_{n+1}(x)-u_{n+1}(y) \bigr) \bigl(\varphi (x)-\varphi (y)\bigr)K(x-y)\,dx\,dy \\ & \quad =\lambda \int _{\varOmega }u_{n}\varphi \,dx+ \int _{\varOmega }u_{n}^{p-1} \varphi \,dx+ \gamma \int _{\varOmega }g\varphi \,dx, \end{aligned}$$
(2.10)
(2.11)

for all \(\varphi \in X_{0}\). Subtract (2.11) from (2.10) and set \(\varphi =(u_{n+1}-w_{n+1})^{+}\). We obtain

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl[\psi _{n+1}(x)-\psi _{n+1}(y)\bigr] \bigl[\psi _{n+1}^{+}(x)- \psi _{n+1}^{+}(y)\bigr]K(x-y)\,dx\,dy\\ & \quad = \lambda \int _{\varOmega }(u_{n}-w_{n}) \psi \,dx+ \int _{\varOmega } \bigl(u_{n}^{p-1}-w_{n}^{p-1} \bigr)\varphi \,dx \le 0, \end{aligned}$$

where \(\psi _{n+1}(x)=u_{n+1}(x)-w_{n+1}(x)\), for all \(x\in \mathbb{R} ^{N}\). Thus \(u_{n+1}\le w_{n+1}\) a.e. in Ω. Similarly we can get \(u_{n}\le u_{n+1}\) and \(w_{n+1}\le w_{n}\).

By Claims 1 and 2 we have

$$ \underline{u}=u_{0}\le u_{1}\le \cdots \le u_{n}\le u_{n+1}\le \cdots \le w_{n+1}\le w_{n}\cdots \le w_{1}\le w_{0}= \overline{u}. $$

Set

$$ u(x)=\lim_{n\rightarrow \infty }u_{n}(x), \qquad w(x)=\lim _{n\rightarrow \infty }w_{n}(x). $$

Clearly, \(u(x)\le w(x)\) a.e. in Ω. Taking \(\varphi = u_{n+1}\) in (2.10) we have

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(u_{n+1}(x)-u_{n+1}(y) \bigr)^{2}K(x-y)\,dx\,dy \\ & \quad = \lambda \int _{\varOmega }u_{n}u_{n+1}\,dx+ \int _{\varOmega }u^{p-1}_{n}u _{n+1}\,dx+\gamma \int _{\varOmega }gu_{n+1}\,dx \\ & \quad \le \lambda \int _{\varOmega }\overline{u}^{2}\,dx+ \int _{\varOmega } \overline{u}^{p}\,dx+\gamma \biggl( \int _{\varOmega }g^{2}\,dx \biggr)^{1/2} \biggl( \int _{\varOmega }\overline{u}^{2}\,dx \biggr)^{1/2}. \end{aligned}$$

This shows \(\{\|u_{n}\|_{X_{0}}\}\) is bounded. So, going if necessary to a subsequence, we can assume that \(u_{n}\rightharpoonup u\) in \(X_{0}\). The Lebesgue’s dominated convergence theorem yields

$$ \int _{\varOmega }u_{n}^{p-1}\varphi \,dx \rightarrow \int _{\varOmega }u^{p-1} \varphi \,dx, \quad \forall \varphi \in X_{0} $$

as \(n\rightarrow \infty \).

Letting \(n\rightarrow \infty \) in (2.10) we have

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(u(x)-u(y)\bigr) \bigl(\varphi (x)-\varphi (y) \bigr)K(x-y)\,dx\,dy\\& \quad = \lambda \int _{\varOmega }u\varphi \,dx + \int _{\varOmega }u^{p-1}\varphi \,dx+ \gamma \int _{\varOmega }g\varphi \,dx, \quad \forall \varphi \in X_{0}. \end{aligned}$$

Similarly we can verify that w is a weak solution of \((P)_{\gamma }\). However, we cannot rule out the possibility that u and w are the same solution. Note that, since \(u\le \varepsilon \phi _{1}\) and \(\phi _{1}\in L^{\infty }(\varOmega )\), we get \(u\in L^{\infty }(\varOmega )\). It is easy to see that \(u(x)>0\) in Ω.

Next we show that u is a minimal solution. Assume that U is any weak solution of \((P)_{\gamma }\). By Lemma 2.2, \(U\wedge \bar{u}:= \min \{U, \bar{u}\}\) is a supersolution of \((P)_{\gamma }\). Using the same method of monotonic iteration we get a positive solution v of \((P)_{\gamma }\) such that \(v\le U\wedge \bar{u}\le \bar{u}\). Using the same argument as proof of Claim 2 above we obtain

$$ u_{n}\le v\le w_{n} \quad \text{for all } n. $$

Passing to the limit we have

$$ u\le v\le w. $$

Consequently, \(u\le v\le U\). This shows that u is a minimal solution. □

Lemma 2.5

For \(\lambda \in [0,\lambda _{1})\) there exists a positive constant \(\gamma ^{*}\) such that \((P)_{\gamma }\) has a positive minimal solution for all \(\gamma \in (0,\gamma ^{*})\), and \((P)_{\gamma }\) has no positive solutions if \(\gamma >\gamma ^{*}\).

Proof

Set

$$ \gamma ^{*}=\sup \bigl\{ \overline{\gamma }>0| (P)_{\gamma } \text{ has at least one positive solution for all } \gamma \in (0,\overline{\gamma }) \bigr\} . $$

Lemma 2.1 and Lemma 2.4 imply that \(\gamma ^{*}\) is well defined.

For any fixed \(\gamma _{0}\in (0,\gamma ^{*})\), we take \(\delta >0\) such that \(\gamma _{0}+\delta <\gamma ^{*}\). Let \(u_{\gamma _{0}+\delta }\) be a positive solution of \((P)_{\gamma _{0}+\delta }\). It is easy to verify that 0 is a subsolution and \(u_{\gamma _{0}+\delta }\) is a supersolution of \((P)_{\gamma _{0}}\). Using the same method of monotonic iteration as that in proof of Lemma 2.4 we find a minimal solution \(u_{\gamma _{0}}\) of \((P)_{\gamma _{0}}\).

By similar arguments we can show there is no positive solution of \((P)_{\gamma }\) for any \(\gamma >\gamma ^{*}\). □

Lemma 2.6

Assume that \(\lambda \in [0,\lambda _{1})\), \(\gamma \in (0,\gamma ^{*})\), where \(\gamma ^{*}\) is the one in Lemma 2.5. Let \(u_{\gamma }\) be the positive minimal solution of \((P)_{\gamma }\). Then

$$\begin{aligned} \tau =&\inf \biggl\{ \int _{\mathbb{R}^{2N}}\bigl(\psi (x)-\psi (y)\bigr)^{2}K(x-y) \,dx\,dy- \lambda \int _{\varOmega }\psi ^{2}\,dx\Bigm| (p-1) \int _{\varOmega }u_{ \gamma }^{p-2}\psi ^{2}\,dx=1, \\ &\psi \in X_{0} \biggr\} \end{aligned}$$
(2.12)

can be attained and \(\tau >1\).

Proof

Clearly, \(0\le \tau <+\infty \). Let \(\{\psi _{n}\}\subset X_{0}\) be a minimizing sequence of (2.12). Then

$$ [\lambda _{1}-\lambda ] \int _{\varOmega }\psi ^{2}_{n}\,dx\le \int _{\mathbb{R}^{2N}}\bigl(\psi _{n}(x)-\psi _{n}(y)\bigr)^{2}K(x-y)\,dx\,dy-\lambda \int _{\varOmega }\psi ^{2}_{n}\,dx=\tau +o(1). $$

So \(|\psi _{n}|_{2}\) is bounded. Since

$$ \int _{\mathbb{R}^{2N}}\bigl(\psi _{n}(x)-\psi _{n}(y)\bigr)K(x-y)\,dx\,dy=\lambda \int _{\varOmega }\psi ^{2}_{n} \,dx+\tau +o(1), $$

we see that \(\|\psi _{n}\|_{X_{0}}\) is bounded. Consequently, we may assume that there is a subsequence, still denoted by \(\psi _{n}\), such that

$$\begin{aligned}& \psi _{n}\rightharpoonup \psi _{0} \quad \text{in } X_{0}, \\& \psi _{n}\rightarrow \psi _{0} \quad \text{in } L^{2}\bigl(\mathbb{R}^{N}\bigr), \\& \psi _{n}\rightarrow \psi _{0} \quad \text{a.e. in } \mathbb{R}^{N}. \end{aligned}$$

Hence, as \(n\rightarrow \infty \),

$$\begin{aligned} \tau =& \liminf_{n\rightarrow \infty } \biggl[ \int _{\mathbb{R}^{2N}}\bigl( \psi _{n}(x)-\psi _{n}(y)\bigr)^{2}K(x-y)\,dx\,dy-\lambda \int _{\varOmega }\psi _{n} ^{2}\,dx \biggr] \\ \ge & \int _{\mathbb{R}^{2N}}\bigl(\psi _{0}(x)-\psi _{0}(y)\bigr)^{2}K(x-y)\,dx\,dy- \lambda \int _{\varOmega }\psi _{0}\,dx. \end{aligned}$$

By the Lebesgue dominated convergence theorem, we have

$$ (p-1) \int _{\varOmega }u_{\gamma }^{p-2}\psi ^{2}_{0}\,dx= \lim_{n\rightarrow \infty }(p-1) \int _{\varOmega }u_{\gamma }^{p-2}\psi ^{2}_{n}\,dx=1. $$

Hence \(\psi _{0}\) reaches τ. Since

$$\begin{aligned} \tau \le& \int _{\mathbb{R}^{2N}}\bigl( \bigl\vert \psi _{0}(x) \bigr\vert - \bigl\vert \psi _{0}(y) \bigr\vert \bigr)^{2}K(x-y)\,dx\,dy- \lambda \int _{\varOmega } \vert \psi _{0} \vert ^{2}\,dx\\ \le& \int _{\mathbb{R}^{2N}}\bigl(\psi _{0}(x)-\psi _{0}(y)\bigr)^{2}K(x-y)\,dx\,dy-\lambda \int _{\varOmega }\psi _{0}^{2}\,dx= \tau , \end{aligned}$$

\(|\psi _{0}|\) also achieves τ. So we can assume \(\psi _{0}\ge 0\) in Ω. It follows from the Lagrange multiplier rule that

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(\psi _{0}(x)-\psi _{0}(y)\bigr) \bigl(\varphi (x)-\varphi (y)\bigr)K(x-y)\,dx\,dy\\& \quad = \lambda \int _{\varOmega }\psi _{0}\varphi \,dx+\tau (p-1) \int _{\varOmega }u _{\gamma }^{p-2}\psi _{0}\varphi \,dx, \quad \forall \varphi \in X_{0}. \end{aligned}$$

We take \(\delta >0\) such that \(\gamma +\delta <\gamma ^{*}\). Set \(\overline{u}=u_{\gamma +\delta }\), where \(u_{\gamma +\delta }\) is a positive solution of \((P)_{\gamma +\delta }\). Then is a supersolution of \((P)_{\gamma }\). Taking \(\varphi =\overline{u}-u_{ \gamma }\) in the equation above we get

$$\begin{aligned}& \begin{gathered}[b] \int _{\mathbb{R}^{2N}}(\psi _{0}(x)-\psi _{0}(y) \bigl[(\overline{u}-u _{\gamma }) (x)-( \overline{u}-u_{\gamma }) (y) \bigr]\,dx\\ \quad =\lambda \int _{ \varOmega }\psi _{0}(\overline{u}-u_{\gamma }) \,dx +\tau (p-1) \int _{\varOmega }u_{\gamma }^{p-2}\psi _{0}(\overline{u}-u_{\gamma })\,dx. \end{gathered} \end{aligned}$$
(2.13)

On the other hand, by the definition of and \(u_{\gamma }\), we have

$$\begin{aligned}& \lambda \int _{\varOmega }(\overline{u}-u_{\gamma })\psi _{0}\,dx+ \int _{ \varOmega }\bigl[\overline{u}^{p-1}-u_{\gamma }^{p-1} \bigr]\psi _{0}\,dx \\& \quad \le \int _{\mathbb{R}^{2N}}(\psi _{0}(x)-\psi _{0}(y) \bigl[(\overline{u}-u _{\gamma }) (x)-( \overline{u}-u_{\gamma }) (y) \bigr]\,dx. \end{aligned}$$
(2.14)

By (2.13) and (2.14) we have

$$\begin{aligned} \tau (p-1) \int _{\varOmega }u_{\gamma }^{p-2}( \overline{u}-u_{\gamma }) \psi _{0}\,dx \ge & \int _{\varOmega }\bigl[\overline{u}^{p-1}-u_{\gamma }^{p-1} \bigr] \psi _{0}\,dx \\ >& (p-1) \int _{\varOmega }u_{\gamma }^{p-2}( \overline{u}-u_{\gamma })\psi _{0}\,dx. \end{aligned}$$

Hence \(\tau >1\). □

Lemma 2.7

There results

$$ \sup_{u_{\gamma }\in \mathcal{S}} \Vert u_{\gamma } \Vert _{X_{0}}< \infty , $$

where

$$ \mathcal{S}=\bigl\{ u_{\gamma }| \gamma \in \bigl(0,\gamma ^{*}\bigr), u_{\gamma } \textit{ is the minimal solution of } (P)_{\gamma }\bigr\} . $$

Proof

For any \(u_{\gamma }\in \mathcal{S}\), from Lemma 2.6 we get

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(u_{\gamma }(x)-u_{\gamma }(y) \bigr)^{2}K(x-y)\,dx\,dy- \lambda \int _{\varOmega }u^{2}_{\gamma }\,dx-(p-1) \int _{\varOmega }u^{p}_{ \gamma }\,dx\\& \quad \ge (\tau -1) (p-1) \int _{\varOmega }u^{p}_{\gamma }\,dx\ge 0. \end{aligned}$$

Consequently,

$$ \int _{\mathbb{R}^{2N}}\bigl(u_{\gamma }(x)-u_{\gamma }(y) \bigr)^{2}K(x-y)\,dx\,dy \ge \lambda \int _{\varOmega }u^{2}_{\gamma }\,dx+(p-1) \int _{\varOmega }u^{p} _{\gamma }\,dx. $$
(2.15)

Clearly,

$$ \int _{\mathbb{R}^{2N}}\bigl(u_{\gamma }(x)-u_{\gamma }(y) \bigr)^{2}K(x-y)\,dx\,dy= \lambda \int _{\varOmega }u^{2}_{\gamma }\,dx+ \int _{\varOmega }u_{\gamma }^{p} \,dx+\gamma \int _{\varOmega }g u_{\gamma }\,dx. $$
(2.16)

By (2.15) and (2.16), we have

$$ \begin{gathered}[b] (p-2) \biggl[ \int _{\mathbb{R}^{2N}}\bigl(u_{\gamma }(x)-u_{\gamma }(y) \bigr)^{2}K(x-y)\,dx\,dy- \lambda \int _{\varOmega }u^{2}_{\gamma }\,dx \biggr] \\ \quad \le (p-1)\gamma \int _{\varOmega }gu_{\gamma }\,dx. \end{gathered} $$
(2.17)

Since

$$ \int _{\mathbb{R}^{2N}}\bigl(u_{\gamma }(x)-u_{\gamma }(y) \bigr)^{2}K(x-y)\,dx\,dy- \lambda \int _{\varOmega }u^{2}_{\gamma }\,dx\ge [\lambda _{1}-\lambda ] \int _{\varOmega }u^{2}_{\gamma }\,dx, $$

we deduce

$$\begin{aligned} \int _{\varOmega }u^{2}_{\gamma }\,dx \le & \frac{(p-1)\gamma }{(p-2)[\lambda _{1}-\lambda ]} \int _{\varOmega }gu_{\gamma }\,dx \\ \le &\frac{(p-1)\gamma }{(p-2)[\lambda _{1}-\lambda ]} \biggl[\frac{1}{2 \delta } \int _{\varOmega }g^{2}\,dx+\frac{\delta }{2} \int _{\varOmega }u^{2} _{\gamma }\,dx \biggr], \end{aligned}$$

for \(\delta >0\) small enough such that

$$ \delta < \frac{2(p-2)[\lambda _{1}-\lambda ]}{(p-1)\gamma }. $$

So there exists a positive constant \(C_{2}\) such that

$$ \int _{\varOmega }u^{2}_{\gamma }\,dx\le C_{2}, $$
(2.18)

where \(C_{2}\) depends only on \(\lambda _{1}\), λ, p, γ, and g.

By (2.17) and (2.18) we have

So there exists a positive constant C independent of γ such that

$$ \Vert u_{\gamma } \Vert _{X_{0}}\le C. $$
(2.19)

 □

Now we prove Theorem 1.1.

Proof of Theorem 1.1

Assume that \(\gamma _{j}\nearrow \gamma ^{*}\) and \(u_{\gamma _{j}}\in \mathcal{S}\). By Lemma 2.7 there is a subsequence, still denoted by \(\{u_{\gamma _{j}}\}\), such that

$$\begin{aligned}& u_{\gamma _{j}}\rightharpoonup u^{*}\quad \text{in } X_{0}, \\& u_{\gamma _{j}}\rightarrow u^{*}\quad \text{in } L^{2}\bigl(\mathbb{R}^{N}\bigr), \\& u_{\gamma _{j}}\rightarrow u^{*}\quad \text{a.e. in } \mathbb{R}^{N}. \end{aligned}$$

It is easy to verify that \(u^{*}\) is a solution of \((P)_{\gamma ^{*}}\). Note that 0 is a subsolution of \((P)_{\gamma }\) for any \(\gamma \ge 0\). So we can use the method of monotone iteration to find a minimal solution. □

Existence of the second positive solution

We introduce the following problem:

$$ \textstyle\begin{cases} (-\Delta )^{s} v=v^{p-1}+a(x)v+h(x,v)& \text{in } \varOmega , \\ v>0& \text{in } \varOmega , \\ v=0 & \text{in } \mathbb{R}^{N}\setminus \varOmega , \end{cases} $$
(3.1)

where \(a(x)=\lambda +(p-1)u_{\gamma }^{p-2}(x)\), and

$$ h(x,v)=\bigl(v+u_{\gamma }(x)\bigr)^{p-1}-u_{\gamma }^{p-1}(x)-v^{p-1}-(p-1)u _{\gamma }^{p-2}v. $$

In order to obtain a second solution of \((P)_{\gamma }\) it suffices to prove (3.1) has a nontrivial solution. Thus \(u_{\gamma }+v\) is a second solution of \((P)_{\gamma }\).

For problem (3.1), we define the energy functional \(J: X_{0} \to \mathbb{R}\) as follows:

$$\begin{aligned} J(v) =&\frac{1}{2} \int _{\mathbb{R}^{2N}}\bigl(v(x)-v(y)\bigr)^{2}K(x-y)\,dx \,dy- \frac{1}{2} \int _{\varOmega }a(x) \bigl(v^{+}\bigr)^{2} \,dx\\ &{}-\frac{1}{p} \int _{\varOmega }\bigl(v ^{+}\bigr)^{p} \,dx- \int _{\varOmega }H\bigl(x,v^{+}\bigr)\,dx, \end{aligned}$$

where \(H(x,v)=\int ^{v}_{0} h(x,t)\,dt\), \(v^{+}=\max \{v,0\}\) denotes the positive part of v. By the maximum principle [2, 16], we know that the nontrivial critical points of energy functional J are the positive solutions of (3.1).

It is easy to see that h satisfies

  1. (i)

    \(\sup \{|h(x,t)|: \text{a.e. } x\in \varOmega , t\le M \}<+ \infty \) for any \(M>0\);

  2. (ii)

    \(\lim_{t\to 0^{+}}\frac{h(x,t)}{t}=0\) uniformly in \(x\in \varOmega \);

  3. (iii)

    \(\lim_{t\to +\infty }\frac{h(x,t)}{t^{p-1}}=0\) uniformly in \(x\in \varOmega \).

The following theorem is a modification of Theorem 3 in [15].

Theorem 3.1

Let \(\lambda \in [0,\lambda _{1})\), \(\gamma \in (0,\gamma ^{*})\), if there exists some \(v_{0}\in X_{0}\setminus \{0\}\) with \(v_{0}\ge 0\) a.e. in \(\mathbb{R}^{N}\), such that

$$ \sup_{t\ge 0}J(tv_{0})< \frac{s}{N}S^{\frac{N}{2s}}_{s}, $$
(3.2)

then problem (3.1) admits a solution.

Since the proof of Theorem 3.1 is nearly same as that of Theorem 3 in [15] (cf. Theorem 2.1 in [1]), we omit it.

In the following, we shall verify the crucial condition (3.2) holds for \(\lambda \in [0,\lambda _{1})\), \(\gamma \in (0,\gamma ^{*})\). To this end, we need some preliminary results.

Consider the following minimization problem:

$$ S_{s}:=\inf_{v\in H^{s}(\mathbb{R}^{N})\setminus \{0\}}\frac{ \int _{\mathbb{R}^{2N}} \vert v(x)-v(y) \vert ^{2}K(x-y)\,dx\,dy}{ (\int _{\mathbb{R}^{N}} \vert v \vert ^{p}\,dx )^{2/p}}. $$

It is well known from [15] that the infimum in the formula above is attained at ũ, where

$$ \tilde{u}(x)=\frac{\kappa }{ (\mu ^{2}+ \vert x-x_{0} \vert ^{2} )^{ \frac{N-2s}{2}}}, \quad x\in \mathbb{R}^{N}, $$
(3.3)

with \(\kappa >0\), \(\mu >0\) and \(x_{0}\in \mathbb{R}^{N}\) fixed constants. Equivalently, the function ū defined as

$$ \bar{u}=\frac{\tilde{u}}{ \Vert \tilde{u} \Vert _{L^{p}(\mathbb{R}^{N})}} $$

is such that

$$ S_{s}= \int _{\mathbb{R}^{2N}} \bigl\vert \bar{u}(x)-\bar{u}(y) \bigr\vert ^{2}K(x-y)\,dx\,dy. $$

The function

$$ u^{*}(x)=\bar{u} \biggl(\frac{x}{S^{1/(2s)}_{s}} \biggr), \quad x\in \mathbb{R}^{N}, $$

is a solution of

$$ (-\Delta )^{s}u= \vert u \vert ^{p-2}u \quad \text{in }\mathbb{R}^{N}. $$
(3.4)

Now, we consider the family of the function \(U_{\varepsilon }\) defined as

$$ U_{\varepsilon }(x)=\varepsilon ^{-(N-2s)/2}u^{*}(x/ \varepsilon ), \quad x\in \mathbb{R}^{N}, $$

for any \(\varepsilon >0\). The function \(U_{\varepsilon }\) is a solution of problem (3.4) and satisfies

$$ \int _{\mathbb{R}^{2N}} \bigl\vert U_{\varepsilon }(x)-U_{\varepsilon }(y) \bigr\vert ^{2}K(x-y)\,dx\,dy= \int _{\mathbb{R}^{N}} \bigl\vert U_{\varepsilon }(x) \bigr\vert ^{p}\,dx=S^{N/(2s)}_{s}. $$
(3.5)

Without loss of generality we may suppose \(0\in \varOmega \). Let us fix \(\rho >0\) such that \(B_{4\rho }\subset \varOmega \) and let \(\eta \in C ^{\infty }\) be such that \(0\le \eta \le 1\) in \(\mathbb{R}^{N}\), \(\eta (x)=1\) if \(|x|<\rho \); \(\eta (x)=0\) if \(|x|\ge 2\rho \). For every \(\varepsilon >0\) we denote by \(u_{\varepsilon }\) the following function:

$$ u_{\varepsilon }(x)=\eta (x)U_{\varepsilon }(x), \quad x\in \mathbb{R}^{N}. $$
(3.6)

In what follows we suppose that up to a translation \(x_{0}=0\) in (3.3). From [15] we have the following estimates:

$$\begin{aligned}& \int _{\mathbb{R}^{2N}} \bigl\vert u_{\varepsilon }(x)-u_{\varepsilon }(y) \bigr\vert ^{2}K(x-y)\,dx\,dy \le S^{N/(2s)}_{s}+O \bigl(\varepsilon ^{N-2s}\bigr), \end{aligned}$$
(3.7)
$$\begin{aligned}& \int _{\mathbb{R}^{N}} \vert u_{\varepsilon } \vert ^{p} \,dx = S^{N/(2s)}_{s}+O\bigl( \varepsilon ^{N}\bigr), \end{aligned}$$
(3.8)
$$\begin{aligned}& \int _{\mathbb{R}^{N}}u^{2}_{\varepsilon }\,dx \ge \textstyle\begin{cases} C_{s}\varepsilon ^{2s}+O(\varepsilon ^{N-2s}),& N>4s, \\ C_{s}\varepsilon ^{2s} \vert \ln \varepsilon \vert +O(\varepsilon ^{2s}),&N=4s, \\ C_{s}\varepsilon ^{N-2s}+O(\varepsilon ^{2s}),&N< 4s, \end{cases}\displaystyle \end{aligned}$$
(3.9)

where \(C_{s}\) is a positive constant depending on s.

Lemma 3.2

Assume that \(\lambda \in [0,\lambda _{1})\), \(\gamma \in (0,\gamma ^{*})\), where \(\gamma ^{*}\) is the one in Lemma 2.5. Let \(u_{\gamma }\) be the positive minimal solution of \((P)_{\gamma }\). Then

$$ \begin{aligned}[b] \hat{\tau }={}&\inf \biggl\{ \int _{\mathbb{R}^{2N}}\bigl(\psi (x)-\psi (y)\bigr)^{2}K(x-y) \,dx\,dy - \int _{\varOmega }a(x)\psi ^{2}\,dx\biggm| \int _{\varOmega }\psi ^{2}\,dx=1,\\ &\ \psi \in X _{0}(\varOmega )\biggr\} \end{aligned} $$
(3.10)

can be attained and \(\hat{\tau }>0\).

Proof

By Lemma 2.6, we have

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(\psi (x)-\psi (y)\bigr)^{2}K(x-y) \,dx\,dy-\lambda \int _{\varOmega }\psi ^{2}\,dx \ge \tau (p-1) \int _{\varOmega }u^{p-2}_{\gamma }\psi ^{2}\,dx, \\& \quad \text{for any } \psi \in X_{0}, \end{aligned}$$

where \(\tau >1\). So,

$$\begin{aligned}& \int _{\mathbb{R}^{2N}}\bigl(\psi (x)-\psi (y)\bigr)^{2}K(x-y) \,dx\,dy- \int _{\varOmega }a(x)\psi ^{2}\,dx \ge (\tau -1) (p-1) \int _{\varOmega }u^{p-2}_{\gamma }\psi ^{2}\,dx, \\& \quad \text{for any } \psi \in X_{0}. \end{aligned}$$

Thus, \(0\le \hat{\tau }<+\infty \). Let \(\{\psi _{n}\}\subset X_{0}\) be a minimizing sequence of (3.10). Then

$$ \int _{\mathbb{R}^{2N}}\bigl(\psi (x)-\psi (y)\bigr)^{2}K(x-y) \,dx\,dy = \int _{\varOmega }a(x)\psi ^{2}_{n}\,dx+ \hat{\tau }+o(1), $$

and \(\int _{\varOmega }\psi ^{2}_{n}\,dx=1\). Since , we have \(\|\psi _{n}\|_{X_{0}}\) is bounded. Consequently, we may assume that there is a subsequence, still denoted by \(\psi _{n}\), such that

$$\begin{aligned}& \psi _{n}\rightharpoonup \psi _{0} \quad \text{in } X_{0}, \\& \psi _{n}\rightarrow \psi _{0} \quad \text{in } L^{2}\bigl(\mathbb{R}^{N}\bigr), \\& \psi _{n}\rightarrow \psi _{0} \quad \text{a.e. in } \mathbb{R}^{N}. \end{aligned}$$

Hence,

$$\begin{aligned} \hat{\tau } =&\lim_{n\to \infty } \biggl( \int _{\mathbb{R}^{2N}}\bigl(\psi _{n}(x)-\psi _{n}(y)\bigr)^{2}K(x-y)\,dx\,dy - \int _{\varOmega }a(x)\psi _{n}^{2}\,dx \biggr) \\ \ge &\lim_{n\to \infty }(\tau -1) (p-1) \int _{\varOmega }u^{p-2}_{\gamma }\psi ^{2}_{n}\,dx \\ =&(\tau -1) (p-1) \int _{\varOmega }u^{p-2}_{\gamma }\psi ^{2}_{0}\,dx>0. \end{aligned}$$

 □

Lemma 3.3

Let \(u_{\varepsilon }\) be given by (3.6). Then there exists a constant \(t_{\varepsilon }>0\) such that

$$ \sup_{t\ge 0}J(tu_{\varepsilon })=J(t_{\varepsilon }u_{\varepsilon }) $$
(3.11)

and

$$ J(t_{\varepsilon }u_{\varepsilon })\le \frac{s}{N}S^{N/(2s)}_{s}- \frac{1}{2} \lambda t_{\varepsilon }^{2} \int _{\varOmega }u_{\varepsilon } ^{2}\,dx- \int _{\varOmega }Q(x,t_{\varepsilon }u_{\varepsilon })\,dx+O \bigl( \varepsilon ^{N-2s}\bigr), $$
(3.12)

where \(Q(x,v)=\int ^{v}_{0} q(x,t)\,dt\) and \(q(x,t)=(t+u_{\gamma }(x))^{p-1}-u _{\gamma }^{p-1}-t^{p-1}\) for \(t\ge 0\).

Proof

Let

$$\begin{aligned} \psi (t) =&J(tu_{\varepsilon }) \\ =&\frac{1}{2}t^{2} \int _{\mathbb{R}^{2N}}\bigl(u_{\varepsilon }(x)-u_{ \varepsilon }(y) \bigr)^{2}K(x-y)\,dx\,dy -\frac{1}{2}t^{2} \int _{\varOmega }a(x)u _{\varepsilon }^{2}\,dx\\ &{}- \frac{1}{p}t^{p} \int _{\varOmega }u^{p}_{\varepsilon }\,dx- \int _{\varOmega }H(x,tu_{\varepsilon })\,dx, \end{aligned}$$

for \(t\ge 0\). Let

$$ \sigma (t)= \int _{\varOmega }H(x,tu_{\varepsilon }). $$

Since for every \(\delta >0\) there exists \(C_{\delta }>0\) such that

$$ \bigl\vert H(x,t) \bigr\vert \le \delta t^{2}+C_{\delta } \vert t \vert ^{p}, $$

for all \(t\ge 0\) and for a.e. \(x\in \varOmega \), we have

$$ \bigl\vert \sigma (t) \bigr\vert \le \delta t^{2} \int _{\varOmega }u^{2}_{\varepsilon }\,dx+C _{\delta }t^{p} \int _{\varOmega }u^{p}_{\varepsilon }\,dx. $$
(3.13)

By Lemma 3.2, there exists \(\hat{\tau }>0\) such that

$$ \int _{\mathbb{R}^{2N}}\bigl(u_{\varepsilon }(x)-u_{\varepsilon }(y) \bigr)^{2}K(x-y)\,dx\,dy- \int _{\varOmega }a(x)u_{\varepsilon }^{2}\,dx\ge \hat{ \tau } \int _{\varOmega }u ^{2}_{\varepsilon }\,dx. $$
(3.14)

By (3.13) and (3.14), there exists a constant \(\alpha >0\) depending on ε such that

$$ \psi (t)=\alpha t^{2}+o \bigl(t^{2}\bigr) $$
(3.15)

for \(\varepsilon <\frac{1}{2}\hat{\tau }\) as \(t\to 0^{+}\).

Next we study ψ for t large. Note that

$$ \psi (t)\le \frac{1}{2}t^{2} \int _{\mathbb{R}^{2N}}\bigl(u_{\varepsilon }(x)-u _{\varepsilon }(y) \bigr)^{2}K(x-y)\,dx\,dy -\frac{1}{p}t^{p} \int _{\varOmega }u ^{p}_{\varepsilon }\,dx $$

and thus \(\psi (t)\to -\infty \) as \(t\to +\infty \). Therefore, we see that there exists \(t_{\varepsilon }>0\) such that

$$ \sup_{t\ge 0}J(tu_{\varepsilon })=J(t_{\varepsilon }u_{\varepsilon }). $$

We show \(\lim_{\varepsilon \to 0} t_{\varepsilon }=1\). Note that for every \(\tilde{\delta }>0\) there exists \(C_{\tilde{\delta }}\) such that

$$ \bigl\vert h(x,t) \bigr\vert \le \tilde{\delta } \vert t \vert ^{p-1}+C_{\tilde{\delta }} \vert t \vert , \qquad \bigl\vert H(x,t) \bigr\vert \le \frac{1}{p}\tilde{\delta } \vert t \vert ^{p}+\frac{1}{2}C_{ \tilde{\delta }} \vert t \vert ^{2} $$
(3.16)

for all \(t\ge 0\) and for a.e. \(x\in \varOmega \).

Clearly,

$$\begin{aligned} 0 =&\frac{d\psi }{\,dt}\bigg|_{t=t_{\varepsilon }} \\ =&t_{\varepsilon } \int _{\mathbb{R}^{2N}} \bigl(u_{\varepsilon }(x)-u_{\varepsilon }(y) \bigr)^{2}K(x-y)\,dx\,dy-t _{\varepsilon } \int _{\varOmega }a(x)u^{2}_{\varepsilon } \,dx \\ &{}-t^{p-1}_{ \varepsilon } \int _{\varOmega }u^{p}_{\varepsilon }\,dx- \int _{\varOmega }h(x,tu _{\varepsilon })u_{\varepsilon }\,dx. \end{aligned}$$
(3.17)

Then, by (3.7) and (3.8), we have

$$\begin{aligned} t^{p-2}_{\varepsilon } =&\frac{\int _{\mathbb{R}^{2N}} (u_{\varepsilon }(x)-u_{\varepsilon }(y))^{2}K(x-y)\,dx\,dy-\lambda \int _{\varOmega }u^{2} _{\varepsilon }\,dx-\frac{1}{t_{\varepsilon }}\int _{\varOmega }q(x,t_{ \varepsilon }u_{\varepsilon })u_{\varepsilon }\,dx}{\int _{\varOmega }u^{p} _{\varepsilon }\,dx} \\ \le &\frac{\int _{\mathbb{R}^{2N}} (u_{\varepsilon }(x)-u_{\varepsilon }(y))^{2}K(x-y)\,dx\,dy}{\int _{\varOmega }u^{p}_{\varepsilon }\,dx} \\ =&\frac{S^{N/(2s)}_{s}+O(\varepsilon ^{N-2s})}{S^{N/(2s)}_{s}+O( \varepsilon ^{N})} =1+O\bigl(\varepsilon ^{N-2s}\bigr) \quad \text{as } \varepsilon \to 0. \end{aligned}$$
(3.18)

On the other hand, by (3.17) and (3.16), we have

$$\begin{aligned} t^{p-2}_{\varepsilon } =&\frac{\int _{\mathbb{R}^{2N}} (u_{\varepsilon }(x)-u_{\varepsilon }(y))^{2}K(x-y)\,dx\,dy-\int _{\varOmega }a(x)u^{2}_{ \varepsilon }\,dx-\frac{1}{t_{\varepsilon }}\int _{\varOmega }h(x,t_{\varepsilon }u_{\varepsilon })u_{\varepsilon }\,dx}{\int _{\varOmega }u^{p}_{\varepsilon }\,dx} \\ \ge &\frac{\int _{\mathbb{R}^{2N}} (u_{\varepsilon }(x)-u_{\varepsilon }(y))^{2}K(x-y)\,dx\,dy-\int _{\varOmega }a(x)u^{2}_{\varepsilon }\,dx- \int _{\varOmega }(\tilde{\delta }t^{p-2}_{\varepsilon }u_{\varepsilon } ^{p}+C_{\tilde{\delta }}u^{2}_{\varepsilon })\,dx}{\int _{\varOmega }u^{p} _{\varepsilon }\,dx} \\ =&\frac{\int _{\mathbb{R}^{2N}} (u_{\varepsilon }(x)-u_{\varepsilon }(y))^{2}K(x-y)\,dx\,dy- \int _{\varOmega }a(x)u^{2}_{\varepsilon }\,dx \,dx}{\int _{\varOmega }u^{p}_{ \varepsilon }\,dx}-\tilde{\delta }t^{p-2}_{\varepsilon }-C_{ \tilde{\delta }} \frac{\int _{\varOmega }u^{2}_{\varepsilon }\,dx}{\int _{ \varOmega }u^{p}_{\varepsilon }\,dx}, \end{aligned}$$

consequently, by (3.7)–(3.9), we get

$$\begin{aligned} t^{p-2}_{\varepsilon } \ge &\frac{1}{1+\tilde{\delta }} \biggl(\frac{ \int _{\mathbb{R}^{2N}} (u_{\varepsilon }(x)-u_{\varepsilon }(y))^{2}K(x-y)\,dx\,dy- \int _{\varOmega }a(x)u^{2}_{\varepsilon }\,dx \,dx}{\int _{\varOmega }u^{p}_{ \varepsilon }\,dx}-C_{\tilde{\delta }}\frac{\int _{\varOmega }u^{2}_{\varepsilon }\,dx}{\int _{\varOmega }u^{p}_{\varepsilon }\,dx} \biggr) \\ \to& \frac{1}{1+ \tilde{\delta }}, \end{aligned}$$
(3.19)

as \(\varepsilon \to 0\). Combining (3.18) and (3.19), we have \(\lim_{\varepsilon \to 0} t_{\varepsilon }=1\).

Let

$$ d_{\varepsilon }=\frac{\int _{\mathbb{R}^{2N}} (u_{\varepsilon }(x)-u _{\varepsilon }(y))^{2}K(x-y)\,dx\,dy}{\int _{\varOmega }u^{p}_{\varepsilon }\,dx}. $$

Since the function \(t\mapsto \frac{1}{2}t^{2}\int _{\mathbb{R}^{2N}}(u _{\varepsilon }(x)-u_{\varepsilon }(y))^{2}K(x-y)\,dx\,dy -\frac{1}{p}t ^{p}\int _{\varOmega }u^{p}_{\varepsilon }\,dx\) is increasing on the interval \([0,d_{\varepsilon }]\), we have, by (3.18),

$$\begin{aligned} J(t_{\varepsilon }u_{\varepsilon }) \le &\frac{1}{2}d_{\varepsilon } ^{2} \int _{\mathbb{R}^{2N}}\bigl(u_{\varepsilon }(x)-u_{\varepsilon }(y) \bigr)^{2}K(x-y)\,dx\,dy -\frac{1}{p}d_{\varepsilon }^{p} \int _{\varOmega }u^{p}_{\varepsilon }\,dx\\ &{}- \frac{1}{2}t_{\varepsilon }^{2} \int _{\varOmega }a(x)u_{\varepsilon }^{2}\,dx- \int _{\varOmega }H(x,t_{\varepsilon }u_{\varepsilon })\,dx \\ =&\frac{s}{N}S^{N/(2s)}_{s}-\frac{1}{2} \lambda t_{\varepsilon }^{2} \int _{\varOmega }u_{\varepsilon }^{2}\,dx- \int _{\varOmega }Q(x,t_{\varepsilon }u_{\varepsilon })\,dx+O \bigl(\varepsilon ^{N-2s}\bigr). \end{aligned}$$

 □

Lemma 3.4

The condition (3.2) holds.

Proof

We consider three cases.

Case 1. \(N>4s\).

By Lemma 3.5 in [4], there exist \(\delta >0\) and \(T>0\) such that

$$ q(x,t)\ge t^{\delta } \quad \text{for } x\in B_{4\rho }, t \ge T. $$

Taking \(\bar{q}(t)=T^{\delta }\chi _{[T,+\infty )}(t)\), then

$$ q(x,t)\ge \bar{q}(t)\ge 0 \quad \text{for }x\in B_{4\rho }, t\ge 0, $$
(3.20)

where \(\chi _{[T,+\infty )}\) denotes the characteristic function of \([T,+\infty )\). Thus

$$ \bar{Q}(t):= \int ^{t}_{0} \bar{q}(s)\,ds\ge T^{1+\delta } \quad \text{for }t\ge 2T. $$

Direct computation yields

$$ \int _{|x|< \rho }\bar{Q}(t_{\varepsilon }u_{\varepsilon })\,dx= \omega _{N-1}\varepsilon ^{N} \int ^{\rho /(S^{1/(2s)}\varepsilon )}_{0}\bar{Q} \biggl(t_{\varepsilon }A \kappa \biggl(\frac{\varepsilon ^{-1}}{\mu ^{2}+t ^{2}} \biggr)^{\frac{N-2s}{2}} \biggr) t^{N-1}\,dt, $$
(3.21)

where \(\omega _{N-1}\) is the area of \(S^{N-1}\), \(A=(\int _{\mathbb{R} ^{N}}\tilde{u}^{p}\,dx)^{1/p}\), \(\kappa , \mu >0\) are constants. By (3.20) and (3.21), we have

$$\begin{aligned} \frac{1}{\varepsilon ^{N-2s}} \int _{|x|< \rho }Q(x,t_{\varepsilon }u_{ \varepsilon })\,dx \ge &\frac{1}{\varepsilon ^{N-2s}} \int _{|x|< \rho } \bar{Q}(t_{\varepsilon }u_{\varepsilon })\,dx \\ \ge &\omega _{N-1}\varepsilon ^{2s} \int ^{C\varepsilon ^{-1/2}}_{0}T^{1+ \delta }t^{N-1} \,dt \\ =&\frac{T^{\delta +1}C^{N}}{N}\varepsilon ^{2s-\frac{N}{2}}, \end{aligned}$$

where \(C>0\) is a some constant such that \(t_{\varepsilon }A\kappa (\frac{\varepsilon ^{-1}}{\mu ^{2}+t^{2}} )^{\frac{N-2s}{2}} \ge 2T\) for all \(t\le C\varepsilon ^{-1/2}\) and ε is small enough. Thus,

$$ \frac{1}{\varepsilon ^{N-2s}} \int _{|x|< \rho }Q(x,t_{\varepsilon }u_{ \varepsilon })\,dx\to + \infty $$

as \(\varepsilon \to 0\) since \(N>4s\).

By (3.12), we have

$$\begin{aligned} J(t_{\varepsilon }u_{\varepsilon }) \le &\frac{s}{N}S^{N/(2s)}_{s}- \frac{1}{2}\lambda t_{\varepsilon }^{2} \int _{\varOmega }u_{\varepsilon } ^{2}\,dx- \int _{\varOmega }Q(x,t_{\varepsilon }u_{\varepsilon })\,dx+O \bigl( \varepsilon ^{N-2s}\bigr) \\ =&\frac{s}{N}S^{N/(2s)}_{s}-\frac{1}{2} \lambda t_{\varepsilon }^{2} \int _{\varOmega }u_{\varepsilon }^{2}\,dx-\varepsilon ^{N-2s} \biggl(\frac{1}{ \varepsilon ^{N-2s}} \int _{|x|< \rho }Q(x,t_{\varepsilon }u_{\varepsilon })\,dx+O(1) \biggr) \\ < &\frac{s}{N}S^{N/(2s)}_{s}, \end{aligned}$$

as \(\varepsilon \to 0\).

Case 2. \(N=4s\).

Clearly, \(p-1=\frac{N+2s}{N-2s}=3\). Note that there exists \(C_{1}>0\) such that

$$ q(x,t)\ge C_{1} t \quad \text{for }x\in B_{4\rho }, t \ge 0. $$

Thus,

$$\begin{aligned}& \frac{1}{\varepsilon ^{N-2s}} \int _{ \vert x \vert < \rho }Q(x,t_{\varepsilon }u_{ \varepsilon })\,dx\\& \quad \ge \frac{1}{\varepsilon ^{N-2s}}\omega _{N-1} \varepsilon ^{N} \int ^{\rho /(S^{1/(2s)}\varepsilon )}_{0}\frac{1}{2}C _{1} \biggl(t_{\varepsilon }A\kappa \biggl(\frac{\varepsilon ^{-1}}{\mu ^{2}+t^{2}} \biggr)^{\frac{N-2s}{2}} \biggr)^{2} t^{N-1}\,dt \\& \quad =\frac{1}{2}C_{1}\omega _{N-1}(t_{\varepsilon }A \kappa )^{2} \int ^{\rho /(S^{1/(2s)}\varepsilon )}_{0} \frac{1}{(\mu ^{2}+t^{2})^{2s}}t^{N-1} \,dt \\& \quad \ge \frac{1}{2}C_{1}\omega _{N-1}(t_{\varepsilon }A \kappa )^{2}4^{-s} \int ^{\rho /(S^{1/(2s)}\varepsilon )}_{\rho /(S^{1/(2s)}\varepsilon ^{1/2})} \frac{1}{t^{4s}}t^{4s-1} \,dt \\& \quad =\frac{1}{4^{1+s}}C_{1}\omega _{N-1}(t_{\varepsilon }A \kappa )^{2} \vert \ln \varepsilon \vert \to +\infty , \end{aligned}$$

as \(\varepsilon \to 0\).

By (3.12), we have

$$\begin{aligned} J(t_{\varepsilon }u_{\varepsilon }) \le &\frac{s}{N}S^{N/(2s)}_{s}- \frac{1}{2}\lambda t_{\varepsilon }^{2} \int _{\varOmega }u_{\varepsilon } ^{2}\,dx- \int _{\varOmega }Q(x,t_{\varepsilon }u_{\varepsilon })\,dx+O \bigl( \varepsilon ^{2s}\bigr) \\ =&\frac{s}{N}S^{N/(2s)}_{s}-\frac{1}{2} \lambda t_{\varepsilon }^{2} \int _{\varOmega }u^{2}_{\varepsilon }\,dx-\varepsilon ^{N-2s} \biggl(\frac{1}{4^{1+s}}C _{1}\omega _{N-1}(t_{\varepsilon }A\kappa )^{2} \vert \ln \varepsilon \vert +O(1) \biggr) \\ < &\frac{s}{N}S^{N/(2s)}_{s}, \end{aligned}$$

as \(\varepsilon \to 0\).

Case 3. \(N<4s\).

Clearly, \(p-1=\frac{N+2s}{N-2s}>3\). By Lemma 3.4 in [4], there exists \(C_{2}>0\) such that

$$ q(x,t)\ge C_{2} t^{p-2} \quad \text{for }x\in B_{4\rho }, t\ge 0. $$

Thus,

$$\begin{aligned}& \frac{1}{\varepsilon ^{N-2s}} \int _{|x|< \rho }Q(x,t_{\varepsilon }u_{ \varepsilon })\,dx \\& \quad \ge \frac{1}{\varepsilon ^{N-2s}}\omega _{N-1} \varepsilon ^{N} \int ^{\rho /(S^{1/(2s)} \varepsilon )}_{0} \frac{1}{p-1}C_{2} \biggl(t_{\varepsilon }A\kappa \biggl(\frac{ \varepsilon ^{-1}}{\mu ^{2}+t^{2}} \biggr)^{\frac{N-2s}{2}} \biggr)^{p-1} t ^{N-1}\,dt \\& \quad =\frac{1}{p-1}C_{2}\omega _{N-1}(t_{\varepsilon }A \kappa )^{p-1} \varepsilon ^{-\frac{N-2s}{2}} \int ^{\rho /(S^{1/(2s)}\varepsilon )} _{0}\frac{1}{(\mu ^{2}+t^{2})^{\frac{N+2s}{2}}}t^{N-1} \,dt \\& \quad \ge \frac{1}{p-1}C_{2}\omega _{N-1}(t_{\varepsilon }A \kappa )^{p-1} \varepsilon ^{-\frac{N-2s}{2}} \int ^{1}_{0}\frac{1}{(\mu ^{2}+t^{2})^{ \frac{N+2s}{2}}}t^{N-1} \,dt \to +\infty , \end{aligned}$$

as \(\varepsilon \to 0\).

By (3.12), we have

$$\begin{aligned} J(t_{\varepsilon }u_{\varepsilon }) \le &\frac{s}{N}S^{N/(2s)}_{s}- \frac{1}{2}\lambda t_{\varepsilon }^{2} \int _{\varOmega }u_{\varepsilon } ^{2}\,dx- \int _{\varOmega }Q(x,t_{\varepsilon }u_{\varepsilon })\,dx+O \bigl( \varepsilon ^{N-2s}\bigr) \\ =&\frac{s}{N}S^{N/(2s)}_{s}-\frac{1}{2} \lambda t_{\varepsilon }^{2} \int _{\varOmega }u^{2}_{\varepsilon }\,dx-\varepsilon ^{N-2s} \biggl(\frac{1}{ \varepsilon ^{N-2s}} \int _{|x|< \rho }Q(x,t_{\varepsilon }u_{\varepsilon })\,dx+O(1) \biggr) \\ < &\frac{s}{N}S^{N/(2s)}_{s}, \end{aligned}$$

as \(\varepsilon \to 0\). □

Proof of Theorem 1.2

By Lemma 3.3 and Theorem 3.1, we see that problem (3.1) has a solution v for \(\lambda \in [0,\lambda _{1})\) and \(\gamma \in (0,\gamma ^{*})\). We can obtain the second solution of \((P)_{\gamma }\) by taking \(u=u_{\gamma }+v\). Combining with Lemma 2.5 we complete our proof. □

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This work is partially supported by NNSFC (No. 11871315).

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YZ contributed the central idea, and wrote the initial draft of the paper. The other authors contributed to refining the ideas, carrying out additional analyses and finalizing this paper. All authors read and approved the final manuscript.

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Wang, F., Zhang, Y. Existence of multiple positive solutions for nonhomogeneous fractional Laplace problems with critical growth. Bound Value Probl 2019, 169 (2019) doi:10.1186/s13661-019-1287-9

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MSC

  • 45G05
  • 35A15
  • 35S15

Keywords

  • Multiple positive solutions
  • Fractional Laplace problems
  • Critical growth
  • Monotonic iteration
  • Variational method