Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent

TL;DR

This paper investigates the existence and multiplicity of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent, linking solutions to the topology of the potential's minimum set and their concentration behavior.

Contribution

It establishes the existence of ground state solutions and relates their number and concentration to the topology of the potential's minimum set.

Findings

Existence of nontrivial nonnegative ground state solutions.

Number of solutions linked to the topology of the potential's minimum set.

Solutions exhibit concentration behavior around minimum points.

Abstract

In present paper, we study the fractional Choquard equation $$\varepsilon^{2s}(-\Delta)^s u+V(x)u=\varepsilon^{\mu-N}(\frac{1}{|x|^\mu}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$ $2^*_s=\frac{2N}{N-2s}$ and $0<\mu<\min\{2s,N-2s\}$. Under suitable assumption on $V$ and $f$, we prove this problem has a nontrivial nonnegative ground state solution. Moreover, we relate the number of nontrivial nonnegative solutions with the topology of the set where the potential attains its minimum values and their's concentration behavior.

Equations

$${\epsilon }^{2s}(-\Delta {)}^{s}u+V(x)u={\epsilon }^{\mu -N}(\frac{1}{|x{|}^{\mu }}\ast F(u))f(u)+|u{|}^{2_s^{\ast }-2}u$$

$${\epsilon }^{2s}(-\Delta {)}^{s}u+V(x)u={\epsilon }^{\mu -N}(\frac{1}{|x{|}^{\mu }}\ast F(u))f(u)+|u{|}^{2_s^{\ast }-2}u,x\in {\mathbb{R}}^N$$

$$(-\Delta {)}^s\Psi (x)=C_{N,s}P.V.{\int }_{{\mathbb{R}}^N}\frac{\Psi (x)-\Psi (y)}{|x-y{|}^{N+2s}}dy,\Psi \in \mathcal{S}({\mathbb{R}}^N),$$

$$i\epsilon \frac{\partial \Psi }{\partial t}={\epsilon }^{2s}(-\Delta {)}^s\Psi +(V(x)+E)\Psi -(K(x)\ast |G(\Psi )|)g(\Psi )(x,t)\in {\mathbb{R}}^N\times \mathbb{R}$$

$$-{\epsilon }^2\Delta u+V(x)u={\epsilon }^{\mu -N}(\frac{1}{|x{|}^{\mu }}\ast F(u))f(u)+|u{|}^{2^{\ast }-2}u\text{in}{\mathbb{R}}^N,$$

$$-\Delta u+u=({\int }_{{\mathbb{R}}^N}\frac{1}{|x{|}^{\mu }}\ast |u{|}^{q}dy)|u{|}^{q-2}u\text{in}{\mathbb{R}}^N.$$

$$-\Delta u+V(\epsilon x+x_0)u=({\int }_{{\mathbb{R}}^N}\frac{G(u(y))}{|x-y{|}^{\mu }}dy)g(u)\text{in}{\mathbb{R}}^N,$$

$$-\Delta u+V(x_0)u=({\int }_{{\mathbb{R}}^N}\frac{G(u(y))}{|x-y{|}^{\mu }}dy)g(u)\text{in}{\mathbb{R}}^N.$$

$$(-\Delta {)}^{s}u+\omega u=({\int }_{{\mathbb{R}}^N}\frac{|u(y){|}^q}{|x-y{|}^{\mu }}dy)|u{|}^{q-2}u\text{in}{\mathbb{R}}^N,$$

$${\epsilon }^{2s}(-\Delta {)}^{s}u+V(x)u={\epsilon }^{\mu -3}({\int }_{{\mathbb{R}}^3}\frac{|u(y){|}^{2_{\mu ,s}^{\ast }}+F(u(y))}{|x-y{|}^{\mu }}dy)(|u{|}^{2_{\mu ,s}^{\ast }-2}u+\frac{1}{2_{\mu ,s}^{\ast }}f(u))\text{in}{\mathbb{R}}^3,$$

$$V_0<\underset{\partial \Omega }{\min }V.$$

$$\underset{\epsilon \to 0}{\lim }V(x_{\epsilon })=V_0,$$

$$H^{s}(\mathbb{R}^{N})=\{u\in L^{2}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\big{(}|\xi|^{2s}|{\mathcal{F}(u)}|^{2}+|{\mathcal{F}(u)}|^{2}\big{)}d\xi<\infty\},$$

$$\|u\|^{2}_{H^{s}(\mathbb{R}^{N})}=\int_{\mathbb{R}^{N}}\big{(}|\xi|^{2s}|{\mathcal{F}(u)}|^{2}+|{\mathcal{F}(u)}|^{2}\big{)}d\xi,$$

$$\|u\|_{\mathcal{D}^{s,2}(\mathbb{R}^{N})}:=\bigg{(}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy\bigg{)}^{\frac{1}{2}}=[u]_{H^{s}(\mathbb{R}^{N})}.$$

$$S_s:=\underset{u\in {\mathcal{D}}^{s,2}({\mathbb{R}}^N)}{\inf }\frac{\Vert u{\Vert }_{{\mathcal{D}}^{s,2}({\mathbb{R}}^N)}^2}{\Vert u{\Vert }_{L^{2_s^{\ast }}({\mathbb{R}}^N)}^2}$$

$$u_{0}(x)=C\big{(}\frac{b}{b^{2}+|x-a|^{2}}\big{)}^{\frac{N-2s}{2}},\ \ x\in\mathbb{R}^{N},$$

$$\mathcal{F}((-\Delta {)}^{s}u)(\xi )=|\xi {|}^{2s}\mathcal{F}(u)(\xi ),\xi \in {\mathbb{R}}^N.$$

$$(-\Delta {)}^{s}u(x)=-\frac{1}{2}C(N,s){\int }_{{\mathbb{R}}^N}\frac{u(x+y)+u(x-y)-2u(x)}{|y{|}^{N+2s}}dy,\forall x\in {\mathbb{R}}^N$$

$$C(N,s)=\bigg{(}\int_{\mathbb{R}^{N}}\frac{(1-cos\xi_{1})}{|\xi|^{N+2s}}d\xi\bigg{)}^{-1},\ \xi=(\xi_{1},\cdots,\xi_{N}).$$

$$[u{]}_{H^s({\mathbb{R}}^N)}^2=\frac{2}{C(N,s)}\Vert (-\Delta {)}^{\frac{s}{2}}u{\Vert }_{L^2({\mathbb{R}}^N)}^2.$$

$$\displaystyle u\longmapsto\bigg{(}\int_{\mathbb{R}^{N}}|u|^{2}dx+\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy\bigg{)}^{\frac{1}{2}};$$

$$\displaystyle u\longmapsto\bigg{(}\int_{\mathbb{R}^{N}}(|\xi|^{2s}|{\mathcal{F}(u)}|^{2}+|{\mathcal{F}(u)}|^{2})d\xi\bigg{)}^{\frac{1}{2}};$$

$$\displaystyle u\longmapsto\bigg{(}\int_{\mathbb{R}^{N}}|u|^{2}dx+\|(-\Delta)^{\frac{s}{2}}u\|^{2}_{L^{2}(\mathbb{R}^{N})}\bigg{)}^{\frac{1}{2}}.$$

$$(-\Delta {)}^{s}u+V(\epsilon x)u=(\frac{1}{|x{|}^{\mu }}\ast F(u))f(u)+|u{|}^{2_s^{\ast }-2}u\text{in}{\mathbb{R}}^N,$$

$$H_{\varepsilon}=\bigg{\{}u\in H^{s}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V(\varepsilon x)u^{2}dx<+\infty\bigg{\}},$$

$$(u,v{)}_{H_{\epsilon }}={\int }_{{\mathbb{R}}^N}(-\Delta {)}^{\frac{s}{2}}u(-\Delta {)}^{\frac{s}{2}}vdx+{\int }_{{\mathbb{R}}^N}V(\epsilon x)uvdx,$$

$$\Vert u{\Vert }_{H_{\epsilon }}^2={\int }_{{\mathbb{R}}^N}|(-\Delta {)}^{\frac{s}{2}}u{|}^{2}dx+{\int }_{{\mathbb{R}}^N}V(\epsilon x)u^{2}dx.$$

$$E_{\epsilon }(u)=\frac{1}{2}\Vert u{\Vert }_{\epsilon }^2-\frac{1}{2}{\int }_{{\mathbb{R}}^N}(\frac{1}{|x{|}^{\mu }}\ast F(u))F(u)dx-\frac{1}{2_s^{\ast }}{\int }_{{\mathbb{R}}^N}|u{|}^{2_s^{\ast }}dx.$$

$$J_{\epsilon }(u)=\frac{1}{2}\Vert u{\Vert }_{\epsilon }^2-\frac{1}{2}{\int }_{{\mathbb{R}}^N}(\frac{1}{|x{|}^{\mu }}\ast F(u^{+}))F(u^{+})dx-\frac{1}{2_s^{\ast }}{\int }_{{\mathbb{R}}^N}|u^{+}{|}^{2_s^{\ast }}dx.$$

$$\begin{array}{rl}⟨J_{\epsilon }^{\prime }(u),\phi ⟩=& \int {\int }_{{\mathbb{R}}^N\times {\mathbb{R}}^N}\frac{u(x)-u(y)}{|x-y{|}^{N+2s}}(\phi (x)-\phi (y))dxdy+{\int }_{{\mathbb{R}}^N}V(\epsilon x)u\phi dx\\ & -{\int }_{{\mathbb{R}}^N}(\frac{1}{|x{|}^{\mu }}\ast F(u^{+}))f(u^{+})\phi dx-{\int }_{{\mathbb{R}}^N}|u^{+}{|}^{2_s^{\ast }-2}u\phi dx.\end{array}$$

Full text

Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent††thanks: This work is supported by NSFC (11361078,11661083,11771385), China.

Shaoxiong Chen, Yue Li, Zhipeng Yang

*Department  of  Mathematics,  Yunnan  Normal  University,  Kunming  650500  P.R. China * Corresponding author:liyue9412@163.com.

Abstract: In present paper, we study the fractional Choquard equation

[TABLE]

where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$ $2^{*}_{s}=\frac{2N}{N-2s}$ and $0<\mu<\min\{2s,N-2s\}$. Under suitable assumption on $V$ and $f$, we prove this problem has a nontrivial nonnegative ground state solution. Moreover, we relate the number of nontrivial nonnegative solutions with the topology of the set where the potential attains its minimum values and their’s concentration behavior.

Key Words: Fractional Choquard equation; Ground state; Lusternik-Schnirelmann theory.

2010 AMS Subject Classification: 35P15, 35P30, 35R11.

1 Introduction and the main results

In this paper, we are interested in the existence, multiplicity and concentration behavior of the semi-classical solutions of the fractional Choquard equation

[TABLE]

where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$ $2^{*}_{s}=\frac{2N}{N-2s},$ $0<\mu<\min\{2s,N-2s\}$ and $F(u)=\int^{t}_{0}f(\tau)d\tau$. The fractional Laplacian $(-\Delta)^{s}$ is defined by

[TABLE]

where $P.V.$ stands for the Cauchy principal value, $C_{N,s}$ is a normalized constant, $\mathcal{S}(\mathbb{R}^{N})$ is the Schwartz space of rapidly decaying functions, $s\in(0,1)$. As $\varepsilon$ goes to zero in (1.1), the existence and asymptotic behavior of the solutions of the singularly perturbed equation (1.1) is known as the semi-classical problem. It was used to describe the transition between of Quantum Mechanics and Classical Mechanics.

Our motivation to study (1.1) mainly comes from the fact that solutions $u(x)$ of (1.1) corresponding to standing wave solutions $\Psi(x,t)=e^{-iEt/\varepsilon}u(x)$ of the following time-dependent fractional Schrödinger equation

[TABLE]

where $i$ is the imaginary unit, $\varepsilon$ is related to the Planck constant. Equations of the type (1.2) was introduced by Laskin (see [22, 23]) and come from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths. With variational methods, this kind equation has been studied widely, we refer to [11, 17, 43] and the references therein.

When $s=1$, the equation (1.1) turns out to be the Choquard equation

[TABLE]

The existence, multiplicity and concentration of solutions for (1.3) has been widely investigated. On one hand, some people have studied the classical problem, namely $\varepsilon=1$ in (1.3). When $V=1$ and $F(u)=\frac{|u|^{q}}{q}$, (1.3) covers in particular the Choquard-Pekar equation

[TABLE]

The case $N=3$, $q=2$ and $\mu=1$ came from Pekar [35] in 1954 to describe the quantum mechanics of a polaron at rest. In 1976 Choquard used (1.4) to describe an electron trapped in its own hole, in a certain approximation to Hartree-Fock theory of one component plasma [24]. In this context (1.4) is also known as the nonlinear Schrödinger-Newton equation. By using critical point theory, Lions [25] obtained the existence of infinitely many radialy symmetric solutions in $H^{1}(\mathbb{R}^{N})$ and Ackermann [1] prove the existence of infinitely many geometrically distinct weak solutions for a general case. For the properties of the ground state solutions, Ma and Zhao [26] proved that every positive solution is radially symmetric and monotone decreasing about some point for the generalized Choquard equation (1.4) with $q\geq 2$. Later, Moroz and Van Schaftingen [28, 29] eliminated this restriction and showed the regularity, positivity and radial symmetry of the ground states for the optimal range of parameters, and also derived that these solutions decay asymptotically at infinity.

On the other hand, some people have focused on the semiclassical problem, namely, $\varepsilon\rightarrow 0$ in (1.3). The question of the existence of semiclassical solutions for the non-local problem (1.3) has been posed in [6]. Note that if $v$ is a solution of (1.3) for $x_{0}\in\mathbb{R}^{N}$, then $u=v(\varepsilon x+x_{0})$ verifies

[TABLE]

which means some convergence of the family of solutions to a solution $u_{0}$ of the limit problem

[TABLE]

For this case when $N=3,\mu=1$ and $G(u)=|u|^{2}$, Wei and Winter [41] constructed families of solutions by a Lyapunov-Schmidt-type reduction when $\inf V>0$. This method of construction depends on the existence, uniqueness and non-degeneracy up to translations of the positive solution of the limiting equation (1.6), which is a difficult problem that has only been fully solved in the case when $N=3,\mu=1$ and $G(u)=|u|^{2}$. Moroz and Van Schaftingen [30] used variational methods to develop a novel non-local penalization technique to show that equation (1.3) with $G(u)=|u|^{q}$ has a family of solutions concentrated at the local minimum of $V$, with $V$ satisfying some additional assumptions at infinity. In addition, Alves and Yang [5] investigated the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method. Very recently, in an interesting paper, Alves et al. [3] study (1.4) with a critical growth, they consider the critical problem with both linear potential and nonlinear potential, and showed the existence, multiplicity and concentration behavior of solutions when the linear potential has a global minimum or maximum.

On the contrary, the results about fractional Choquard equation (1.1) are relatively few. Recently, d’Avenia, Siciliano and Squassina [15] studied the existence, regularity and asymptotic of the solutions for the following fractional Choquard equation

[TABLE]

where $\omega>0$, $\frac{2N-\mu}{N}<q<\frac{2N-\mu}{N-2s}$. Shen, Gao and Yang [37] obtain the existence of ground states for (1.7) with general nonlinearities by using variational methods. Chen and Liu [13] studied (1.7) with nonconstant linear potential and proved the existence of ground states without any symmetry property. For critical problem, Wang and Xiang [39] obtain the existence of infinitely many nontrivial solutions and the Brezis-Nirenberg type results can be founded in [33]. For other existence results we refer to [8, 9, 19, 20, 27, 40, 46] and the references therein.

For the concentration behavior of solutions, we note that the only works concerning the concentration behavior of solutions come from [42, 44]. Assuming the global condition on $V\in C(\mathbb{R}^{N},\mathbb{R})$:

• $(V_{0})$

$0<V_{0}:=\inf\limits_{x\in\mathbb{R}^{N}}V(x)<\liminf\limits_{|x|\rightarrow\infty}V(x):=V_{\infty}<+\infty$,

which is firstly introduced by Rabinowitz [36] in the study of the nonlinear Schrödinger equations. By using the method of Nehari manifold developed by Szulkin and Weth [38], Zhang, Wang and Zhang in [44] obtained the multiplicity and concentration of positive solutions for the following fractional Choquard equation

[TABLE]

where $\varepsilon>0$, $0<\mu<3$, $F$ is the primitive function of $f$. Different to the global condition $(V_{0})$, Yang in [42] establish the existence and concentration of positive solutions for the fractional Choquard equation (1.8) when the potential function $V\in C(\mathbb{R}^{3},\mathbb{R})$ satisfies the following local conditions [16]:

• $(V_{1})$

There is constant $V_{0}>0$ such that $V_{0}=\inf\limits_{x\in\mathbb{R}^{3}}V(x)$.

• $(V_{2})$

There is a bounded domain $\Omega$ such that

[TABLE]

Note that in (1.8), the critical term is involved in the convolution-type nonlinearity, which is totally different from our problem (1.1). It is natural to ask how about the concentration behavior of solutions of (1.1) as $\varepsilon\to 0^{+}$? And how about the influence of the potential on the multiplicity of solutions? However, to the best of our knowledge, it seems that these two problems were not considered in literatures before. In this paper, we are concerned with the multiplicity and concentration property of nontrivial nonnegative solutions to (1.1), and we will give some answers to the above questions.

Concerning the continuous function $f\in C(\mathbb{R},\mathbb{R})$, we assume that $f(t)=0$ for $t<0$ and satisfies the following conditions:

• $(f_{1})$

$\lim\limits_{t\rightarrow 0}\frac{f(t)}{t}=0$.

• $(f_{2})$

$\exists$ $q\in(\frac{2N-\mu}{N},\frac{2N-\mu}{N-2s})$ such that $\lim\limits_{t\rightarrow\infty}\frac{f(t)}{t^{q-1}}=0.$

• $(f_{3})$

$\frac{f(t)}{t}$ is increasing for every $t>0$.

• $(f_{4})$

$\exists$ $\sigma\in(q_{{}_{N}},\frac{2N-\mu}{N-2s}),$ $C>0$ $s.t.$ $f(t)\geq ct^{\sigma-1}$ for all $t\in\mathbb{R}^{+},$ where $q_{{}_{N}}=\max\{\frac{2N-2s}{N-2s},\frac{N+2s}{N-2s}\}.$

Then we state our main result as follows.

Theorem 1.1

Suppose $(V_{0})$ hold and $f$ satisfies $(f_{1})-(f_{4})$. Then there exists an $\varepsilon^{*}>0$ such that for any $\varepsilon\in(0,\varepsilon^{*}),$ the problem (1.1) possesses a nontrivial nonnegative ground state solution.

In order to describe the multiplicity, we first recall that, if $Y$ is a closed subset of a topological space $X$, the Ljusternik-Schnirelmann category $cat_{X}Y$ is the least number of closed and contractible sets in $X$ which cover $Y$. Then we have our second result as follows.

Theorem 1.2

Suppose $(V_{0})$ hold and $f$ satisfies $(f_{1})-(f_{4})$. Then for any $\delta>0$, there exists $\varepsilon_{\delta}>0$ such that for any $\varepsilon\in(0,\varepsilon_{\delta}),$ the problem 1.1 has at least $cat_{\Lambda_{\delta}}(\Lambda)$ nontrivial nonnegative solutions. Moreover, if $u_{\varepsilon}$ denotes one of these solutions and $x_{\varepsilon}\in{\mathbb{R}^{N}}$ is its global maximum, then

[TABLE]

where $\Lambda:=\{x\in\mathbb{R}^{N}:V(x)=V_{0}\}$ and $\Lambda_{\delta}:=\{x\in\mathbb{R}^{N}:d(x,\Lambda)\leq\delta\}.$

We shall use the method of Nehari manifold, concentration compactness principle and category theory to prove the main results. There are some difficulties in proving our theorems. The first difficulty is that the nonlinearity $f$ is only continuous, we need to prove the new Brezis-Lieb type Lemma for this kind of nonlinearity. The second one is the lack of compactness of the embedding of $H^{s}(\mathbb{R}^{3})$ into the space $L^{2^{*}_{s}}(\mathbb{R}^{3})$. We shall borrow the idea in [3, 12] to deal with the difficulties brought by the critical exponent. However, we require some new estimates, which are complicated because of the appearance of fractional Laplacian and the convolution-type nonlinearity.

This paper is organized as follows. In section 2, besides describing the functional setting to study problem (1.1), we give some preliminary Lemmas which will be used later. In section 3, we prove problem (1.1) has a ground state solution. Finally, we show the multiple of nontrivial nonnegative solutions and investigate its concentration behavior, which completes the proof Theorem 1.1.

Notation. In this paper we make use of the following notations.

• $\bullet$

For any $R>0$ and for any $x\in\mathbb{R}^{N}$, $B_{R}(x)$ denotes the ball of radius $R$ centered at $x$.

• $\bullet$

$L^{p}(\mathbb{R}^{N})$, $1\leq p<+\infty$ denotes the Lebesgue space with the norm $\|u\|_{p}=\|u\|_{L^{p}(\mathbb{R}^{N})}=(\int_{\mathbb{R}^{N}}|u|^{p}dx)^{\frac{1}{p}}$.

• $\bullet$

The letters $C,C_{i}$ stand for positive constants (possibly different from line to line).

• $\bullet$

”$\rightarrow$” for the strong convergence and ”$\rightharpoonup$” for the weak convergence.

• $\bullet$

$u^{+}=\max\{u,0\}$ and $u^{-}=\min\{u,0\}$ denote the positive part and the negative part of a function $u$, respectively.

2 Functional Setting

Firstly, fractional Sobolev spaces are the convenient setting for our problem, so we will give some skrtchs of the fractional order Sobolev spaces and the complete introduction can be found in [17]. We recall that, for any $s\in(0,1)$, the fractional Sobolev space $H^{s}(\mathbb{R}^{N})=W^{s,2}(\mathbb{R}^{N})$ is defined as follows:

[TABLE]

whose norm is defined as

[TABLE]

where $\mathcal{F}$ denotes the Fourier transform. We also define the homogeneous fractional Sobolev space $\mathcal{D}^{s,2}(\mathbb{R}^{N})$ as the completion of $\mathcal{C}_{0}^{\infty}(\mathbb{R}^{N})$ with respect to the norm

[TABLE]

The embedding $\mathcal{D}^{s,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{*}_{s}}(\mathbb{R}^{N})$ is continuous and for any $s\in(0,1)$, there exists a best constant $S_{s}>0$ such that

[TABLE]

According to [14], $S_{s}$ is attained by

[TABLE]

where $C\in\mathbb{R}$, $b>0$ and $a\in\mathbb{R}^{N}$ are fixed parameters.

The fractional laplacian, $(-\Delta)^{s}u$, of a smooth function $u:\mathbb{R}^{N}\rightarrow\mathbb{R}$, is defined by

[TABLE]

Also $(-\Delta)^{s}u$ can be equivalently represented [17] as

[TABLE]

where

[TABLE]

Also, by the Plancherel formular in Fourier analysis, we have

[TABLE]

For convenience, we will omit the normalization constant in the following. As a consequence, the norms on $H^{s}(\mathbb{R}^{N})$ defined below

[TABLE]

are equivalent.

Making the change of variable $x\mapsto\varepsilon x$, we can rewrite the equation (1.1) as the following equivalent form

[TABLE]

If $u$ is a solution of the equation (2.2), then $v(x):=u(\frac{x}{\varepsilon})$ is a solution of the equation (1.1). Thus, to study the equation (1.1), it suffices to study the equation (2.2). In view of the presence of potential $V(x)$, we introduce the subspace

[TABLE]

which is a Hilbert space equipped with the inner product

[TABLE]

and the norm

[TABLE]

We denote $\|\cdot\|_{H_{\varepsilon}}$ by $\|\cdot\|_{\varepsilon}$ in the sequel for convenience. The energy functional corresponding to equation (2.2) is

[TABLE]

Since we are interested in the nontrivial nonnegative solutions, we consider the following functional

[TABLE]

Moreover, $J_{\varepsilon}(u)\in C^{1}(H^{s},\mathbb{R}^{N}),$

[TABLE]

We collect the following useful result.

Lemma 2.1

Let $s\in(0,1)$ and $N>2s.$ Then there exists a sharp constant $C_{\ast}=C(N,s)>0$ such that for any $u\in H^{s}(\mathbb{R}^{N})$

[TABLE]

Moreover $H^{s}(\mathbb{R}^{N})$ is continuously embedded in $L^{q}(\mathbb{R}^{N})$ for any $q\in[2,2^{*}_{s}]$ and compactly in $L^{q}_{loc}(\mathbb{R}^{N})$ for any $q\in[2,2^{*}_{s}).$

Lemma 2.2

Let $N>2s,$ If $\{u_{n}\}$ is a bounded sequence in $H^{s}(\mathbb{R}^{N})$ and if

[TABLE]

where $R>0,$ then $u_{n}\rightarrow 0$ in $L^{t}(\mathbb{R}^{N})$ for all $t\in[2,2^{*}_{s}).$

Lemma 2.3

Let $t,r>1$ and $0<\mu<N$ such that $\frac{1}{r}+\frac{\mu}{N}+\frac{1}{t}=2.$ Let $f\in L^{r}(\mathbb{R}^{N})$ and $h\in L^{t}(\mathbb{R}^{N}).$ Then there exists a sharp constant $C(r,N,\mu,t)>0,$ independent of $f\text{and }h,$ such that

[TABLE]

Lemma 2.4

The space $H_{\varepsilon}$ is continuously embedded into $H^{s}(\mathbb{R}^{N}).$ Therefore, $H_{\varepsilon}$ is continously embedded into $L^{r}(\mathbb{R}^{N})$ for any $r\in[2,2^{*}_{s}]$ and compactly embedded into $L^{r}_{loc}(\mathbb{R}^{N})$ for any $r\in[2,2^{*}_{s}).$

Lemma 2.5

[34*]*Let $u\in\mathcal{D}^{s,2}(\mathbb{R}^{N})$, $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N})$ and for each $r>0,~{}\varphi_{r}(x)=\varphi(\frac{x}{r})$. Then

[TABLE]

If, in addition, $\varphi\equiv 1$ in a neighbourhood of the origin, then

[TABLE]

3 Ground state solution

Lemma 3.1

$J_{\varepsilon}$* has a mountain pass geometry, that is*

• $(i)$

There exists $\alpha,\rho>0$ such that $J_{\varepsilon}(u)\geq\alpha$ for any $u\in H_{\varepsilon}$ which $\|u\|_{\varepsilon}=\rho.$

• $(ii)$

There exists $e\in H_{\varepsilon}$ with $\|e\|_{\varepsilon}>\rho$ such that $J_{\varepsilon}(e)<0.$

Proof:  In order to show this, we argue as in Lemma2.2 in [7]. From $(f_{1})$ and $(f_{2}),$ it follows that for any $\xi>0$ there exists $C_{\xi}>0$ such that

[TABLE]

By (3.1) and Lemma 2.3, we get

[TABLE]

where $t=\frac{2N}{2N-\mu}.$ Since $q\in(\frac{2N-\mu}{N},\frac{2N-\mu}{N-2s}),$ we can see that $tq\in(2,2^{*}_{s}),$ and from Lemma 2.4, we have

[TABLE]

Taking into account (3.2) and (3.3) we can deduce that

[TABLE]

As a consequence

[TABLE]

We can see that $(i)$ holds.

Fix a positive function $u_{0}\in H_{\varepsilon}(\mathbb{R}^{N})\setminus\{0\}$ and $u_{0}>0,$ we set

[TABLE]

By $(f_{3}),$ we have

[TABLE]

Hence,

[TABLE]

Integrating (3.5) on $[1,t\|u_{0}\|_{\varepsilon}]$ with $t>\frac{1}{\|u_{0}\|_{\varepsilon}},$ we find

[TABLE]

which gives

[TABLE]

Therefore, we have

[TABLE]

for $t>\frac{1}{\|u_{0}\|_{\varepsilon}}.$ Taking $e=tu_{0}$ with $t$ sufficiently large, we can see that $(ii)$ holds.    

Let us denote by $\mathcal{S}_{\varepsilon}$ the unitary sphere in $H_{\varepsilon}.$

Lemma 3.2

For each $u\in X^{+}_{\varepsilon}:=\{u\in H_{\varepsilon}:u^{+}(x)\neq 0\}$ and $t>0,$ set $h_{u}(t):=J_{\varepsilon}(tu).$

• $(i)$

Then there exists an unique $t_{u}>0$ such that $h_{u}(t_{u})=\max\limits_{t\geq 0}h_{u}(t)=\max\limits_{t\geq 0}J_{\varepsilon}(tu),$ $h^{\prime}_{u}(t_{u})=0,$ $h^{\prime}_{u}(t)>0$ in $(0,t_{u}),$ $h^{\prime}_{u}(t)<0$ in $(t_{u},+\infty)$ and $tu\in\mathcal{N}_{\varepsilon}$ if and only if $t=t_{u},$ where $\mathcal{N}_{\varepsilon}=\{u\in X^{+}_{\varepsilon}:\langle J^{\prime}_{\varepsilon}(u),u\rangle=0\}.$

• $(ii)$

There is $\kappa>0$ independent on $u,$ such that $t_{u}\geq\kappa$ for all $u\in\mathcal{S}_{\varepsilon}.$ Moreover, for any compact set $E\subset\mathcal{S}_{\varepsilon},$ there is a $C_{E}>0$ such that $t_{u}\leq C_{E}$ for all $u\in E.$

Proof:  $(i)$ For every $u\in X^{+}_{\varepsilon},$ from Lemma (3.1) we know that $h_{u}(0)=0,$ $h_{u}(t)>0$ for $t>0$ small enough and $\lim\limits_{t\rightarrow+\infty}h_{u}(t)=-\infty.$ Hence, there exists a $t_{u}>0$ such that $h_{u}(t_{u})=\max\limits_{t\geq 0}h_{u}(t)$ and $h^{\prime}_{u}(t_{u})=0.$ Notice that

[TABLE]

From $(f_{3})$ we know $t\mapsto f(t)$ and $t\mapsto\frac{F(t)}{t}$ are increasing for all $t>0.$ Hence, we get the uniqueness of a such $t_{u}$ and $(i)$ is completed.

$(ii)$ Let $u\in\mathcal{S}_{\varepsilon}.$ By $t_{u}u\in\mathcal{N}_{\varepsilon}$ and (3.4) we have

[TABLE]

So, there exists $\kappa>0$ independent of $u$, such that $t_{u}\geq\kappa.$ Let, $\alpha\in(2,2^{*}_{s}),$ $\alpha\leq 4$ then $\frac{2}{\alpha}\geq\frac{1}{2}.$ We can infer that

[TABLE]

For any $v\in\mathcal{N}_{\varepsilon},$ we have

[TABLE]

If $E\subset\mathcal{S}_{\varepsilon}$ is a compact set and $u_{n}\subset E$ such that $t_{u_{n}}\rightarrow\infty,$ up to subset $u_{n}\rightarrow u$ in $H_{\varepsilon}$ and $J_{\varepsilon}(t_{u_{n}}u_{n})\rightarrow-\infty.$ Taking $v_{n}=t_{u_{n}}u_{n}\in\mathcal{N}_{\varepsilon}$ in (3.6), we can see that

[TABLE]

which gives a contradiction.

Define the mappings $\hat{n}_{\varepsilon}:H_{\varepsilon}\backslash\{0\}\rightarrow\mathcal{N}_{\varepsilon}$ and $n_{\varepsilon}:=\mathcal{S}_{\varepsilon}\rightarrow\mathcal{N}_{\varepsilon}$ by set

[TABLE]

We can apply [38, Proposition8, Proposition9 and Corollary10 ] to deduce the follow Lemma.

Lemma 3.3

Suppose that $(V_{0})$ and $(f_{1})-(f_{4}),$ then

• $(a)$

The mapping $\hat{n}_{\varepsilon}$ is continuous and $n_{\varepsilon}$ is a homeomorphism between $\mathcal{S}_{\varepsilon}$ and $\mathcal{N}_{\varepsilon}.$ Moreover, $n_{\varepsilon}^{-1}(u)=\frac{u}{\|u\|_{\varepsilon}}.$

• $(b)$

We define the maps $\hat{\psi}_{\varepsilon}:H_{\varepsilon}\backslash\{0\}\rightarrow\mathbb{R}$ by $\hat{\psi}_{\varepsilon}(u):=J_{\varepsilon}(\hat{n}_{\varepsilon}(u)).$ Then $\hat{\psi}_{\varepsilon}\in C^{1}(H_{\varepsilon}\backslash\{0\},\mathbb{R})$ and

[TABLE]

for every $u\in H_{\varepsilon}\backslash\{0\}$ and $v\in H_{\varepsilon}.$

• $(c)$

We define the maps $\psi:\mathcal{S}_{\varepsilon}\rightarrow\mathbb{R}$ by $\psi_{\varepsilon}:=\hat{\psi}|_{\mathcal{S}_{\varepsilon}}.$ Then $\psi_{\varepsilon}\in C^{1}(\mathcal{S}_{\varepsilon},\mathbb{R})$ and $\langle\psi^{\prime}_{\varepsilon}(u),v\rangle=\|n_{\varepsilon}(u)\|_{\varepsilon}\langle J_{\varepsilon}^{\prime}(n_{\varepsilon}(u)),v\rangle$ for any $v\in T_{u}\mathcal{S}_{\varepsilon}.$

• $(d)$

If $\{u_{n}\}$ is a $(PS)_{d}$ sequence for $\psi_{\varepsilon},$ then $\{n_{\varepsilon}(u_{n})\}$ is a $(PS)_{d}$ sequence for $J_{\varepsilon}.$ Moreover, if $\{u_{n}\}\subset\mathcal{N}_{\varepsilon}$ is a bounded $(PS)_{d}$ sequence for $\psi_{\varepsilon},$ then $\{n_{\varepsilon}^{-1}(u_{n})\}$ is a $(PS)_{d}$ sequence for the functional $\psi_{\varepsilon}.$

• $(e)$

$u$* is a critical point of $\psi_{\varepsilon}$ if and only if $n_{\varepsilon}(u)$ is a nontrivial critical point for $J_{\varepsilon}.$ Moreover, the corresponding critical values coincide and*

[TABLE]

Remark 3.1

As in [38], we have the following minimax characterization:

[TABLE]

Recall that $\{u_{n}\}\subset H_{\varepsilon}$ is called a $(C)_{d}$ sequence of $J_{\varepsilon}$ if $J_{\varepsilon}(u_{n})\rightarrow d$ and $(1+\|u_{n}\|_{\varepsilon})J^{\prime}_{\varepsilon}(u_{n})\rightarrow 0,$ where $d>0$ $J_{\varepsilon}$ satisfies the $(C)_{d}$ condition if every $(C)_{d}$ sequence of $J_{\varepsilon}$ has a convergent subsequence. And $J_{\varepsilon}$ satisfies the $(PS)_{d}$ condition if $J_{\varepsilon}$ satisfies the $(C)_{d}$ condition. Next, we give some properties of $(PS)_{d}$ sequence of $J_{\varepsilon}.$

Lemma 3.4

Let $\{u_{n}\}\subset H_{\varepsilon}$ is a $(PS)_{d}$ sequence of $J_{\varepsilon},$ then $\{u_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N})$ and $\{\|u_{n}^{-}\|_{\varepsilon}\}=o_{n}(1).$

Proof:  Let, $\alpha\in(2,2^{*}_{s}),$ $\alpha\leq 4$ then $\frac{2}{\alpha}\geq\frac{1}{2}.$ We can infer that

[TABLE]

Since $\{u_{n}\}$ is a $(PS)_{d}$ sequence of $J_{\varepsilon},$ we have

[TABLE]

Therefore, we get that the sequence $\{u_{n}\}$ is bounded in $H_{\varepsilon}.$ Next, we prove that ${\|u_{n}^{-}\|}=o_{n}(1).$ Since $\langle J^{\prime}_{\varepsilon}(u_{n}),u_{n}^{-}\rangle=o_{n}(1),$ by using $f(t)=0$ for $t\leq 0$ and $(x-y)(x^{-}-y^{-})\geq|x^{-}-y^{-}|^{2}$ where $x^{-}=\min\{x,0\},$ we can deduce that

[TABLE]

Therefore, we complete our proof.

Lemma 3.5

There exists a constant $r>0$ such that $\|u\|_{\varepsilon}\geq r$ for all $\varepsilon\geq 0$ and $u\in\mathcal{N}_{\varepsilon}$

Proof:  By using Lemma 2.3 and $(f_{1})-(f_{2}),$ we can see that for any $u\in\mathcal{N}_{\varepsilon}$

[TABLE]

then, there exists $r>0$ such that

[TABLE]

Hence, we deduce to the lemma holds.

When $V\equiv 1,$ then $H^{s}(\mathbb{R}^{N})=H_{\varepsilon}(\mathbb{R}^{N}).$ For $\tau>0$ and $u\in H^{s}(\mathbb{R}^{N}),$ let

[TABLE]

[TABLE]

For $m_{\tau},$ there also holds

[TABLE]

where $\Gamma=\{\gamma\in C([0,1],H^{s}(\mathbb{R}^{N})):\gamma(0)=0,\,\,\ I_{\tau}(\gamma(1))<0\}.$

Lemma 3.6

For any $\tau>0,$ there exists $u\in H^{s}(\mathbb{R}^{N})$ with $u^{+}\neq 0$ such that

[TABLE]

where $S:=\inf\limits_{u\in\mathcal{D}^{s,2}(\mathbb{R}^{N})}\frac{\|u\|^{2}_{\mathcal{D}^{s,2}(\mathbb{R}^{N})}}{\|u\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}^{2}}.$

Proof:  Let $\varphi\in C^{\infty}_{0}(\mathbb{R}^{N})$ be such that $\varphi=1$ in $B_{\delta},$ $\varphi(x)=0$ in $\mathbb{R}^{N}\setminus B_{2\delta}.$ Denote

[TABLE]

where $u^{*}(x)=\frac{\tilde{u}(x/{S^{\frac{1}{2s}}})}{\|\tilde{u}\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}},$ $\tilde{u}(x/{S^{\frac{1}{2s}}})=\frac{\alpha}{(1+|x/{S^{\frac{1}{2s}}}|^{2})^{\frac{N-2s}{2}}}$ with $\alpha>0.$ We define

[TABLE]

then $u_{\varepsilon}\in H_{\varepsilon}.$ From [17] and [33], we have the following estimations

[TABLE]

[TABLE]

[TABLE]

A standard argument shows that for any $u_{\varepsilon},$ there exists a unique $t_{\varepsilon}$ such that $t_{\varepsilon}u_{\varepsilon}\in\mathcal{M}_{\tau}$ and $I_{\tau}(t_{\varepsilon}u_{\varepsilon})=\max\limits_{t\geq 0}I_{\tau}(tu_{\varepsilon}).$ As a consequence $m_{\tau}\leq I_{\tau}(t_{\varepsilon}u_{\varepsilon})$ and

[TABLE]

As a consequence $t_{\varepsilon}\geq t_{0},$ where $t_{0}>0$ is independent of $\varepsilon.$ Now, we estimate the convolution term. For $\varepsilon>0$ small enough, we have

[TABLE]

Set $g(t):=\frac{t^{2}}{2}([u_{\varepsilon}]_{H^{s}(\mathbb{R}^{N})}^{2}+\tau\int_{\mathbb{R}^{N}}u_{\varepsilon}^{2}dx)-\frac{t^{2^{*}_{s}}}{2^{*}_{s}}\int_{\mathbb{R}^{N}}|u_{\varepsilon}|^{2^{*}_{s}}dx.$ If $N>4s,$ by a simple calculation, we get

[TABLE]

Nothing that $\sigma\in[q_{{}_{N}},\frac{2N-\mu}{N-2s}),$ for $\varepsilon>0$ small enough, using (3.11),(3.12) we can check

[TABLE]

In similar way, we can check $N=4s$ and $N<4s.$

Lemma 3.7

Let $\{u_{n}\}\subset H_{\varepsilon}$ be a $(PS)_{d}$ sequence of $J_{\varepsilon}$ with $d<\frac{s}{N}S^{\frac{N}{2s}}$ and $u_{n}\rightharpoonup 0$ in $H_{\varepsilon}.$ Then one of the following conclusions holds:

• $(a)$

$u_{n}\rightarrow 0$* in $H_{\varepsilon};$*

• $(b)$

There exists a sequence $\{y_{n}\}\subset\mathbb{R}^{N}$ and positive constants $r,\beta,$ such that

[TABLE]

Proof:  If $(b)$ does not occur, then for all $R>0,$ up to a subsequence

[TABLE]

Since we know that $\{u_{n}\}$ is bounded in $H_{\varepsilon},$ we can use Lemma 2.2 to deduce that $u_{n}\rightarrow 0$ in $L^{r}(\mathbb{R}^{N})$ for any $r\in(2,2^{*}_{s}).$ So, apply Hardy-Littlewood-Sobolev inequality, we know that

[TABLE]

Taking into account $\langle J^{\prime}_{\varepsilon}(u_{n}),u_{n}\rangle=o_{n}(1)$ we can infer that

[TABLE]

Since $\{u_{n}\}$ is bounded, up to a subsequence, we have

[TABLE]

If $l>0,$ then

[TABLE]

as $n\rightarrow\infty,$ hence $l\geq S^{\frac{N}{2s}}.$ Consequently, by (3.13), we have

[TABLE]

a contradiction, hence $l=0.$ Consequently, by the boundedness of $\{u_{n}\}$ in $H_{\varepsilon},$ we have $u_{n}\rightarrow 0$ in $H_{\varepsilon},$ so $(a)$ holds. This completes the proof.

Lemma 3.8

Let $\{u_{n}\}\subset H_{\varepsilon}$ be a $(PS)_{d}$ sequence of $J_{\varepsilon}$ with $d<m_{{}_{V_{\infty}}}$ and $u_{n}\rightharpoonup 0$ in $H_{\varepsilon}.$ Then $u_{n}\rightarrow 0$ in $H_{\varepsilon}.$

Proof:  By Lemma 3.4 we can assume $u_{n}\geq 0.$ For any subsequence of $\{u_{n}\}$ still denoted by $\{u_{n}\}.$ Since $u_{n}\rightharpoonup 0$ in $H_{\varepsilon},$ up to a subsequence, we can assume

[TABLE]

If $u_{n}\nrightarrow 0$ in $H_{\varepsilon},$ by Lemma 3.2 we know for any $\{t_{n}\}\subset(0,+\infty)$ $s.t.$ $\{t_{n}u_{n}\}\subset\mathcal{N}_{V_{\infty}}.$

Claim $\limsup\limits_{n\rightarrow\infty}\leq 1.$ If does not occur for any $\delta>0,$ consider any subsequence of $\{t_{n}\}$ and satisfies the following

[TABLE]

Since $\{u_{n}\}$ is a $(PS)_{d}$ sequence of $J_{\varepsilon},$ we can see that

[TABLE]

We observe that $\{t_{n}u_{n}\}\subset\mathcal{N}_{V_{\infty}},$ we have

[TABLE]

Taking into account (3.14) and (3.15) we can deduce that

[TABLE]

By $(V_{0})$, for any $\xi>0$ there exists $R(\xi):=R>0$ such that

[TABLE]

Notice that $u_{n}\rightarrow 0$ in $L^{2}(B_{R}(0))$ and the boundedness of $\{u_{n}\}$ in $H_{\varepsilon},$ we get

[TABLE]

If $u_{n}\nrightarrow 0,$ $\exists$ $\{y_{n}\}\subset\mathbb{R}^{N},$ $r,\delta>0$ such that

[TABLE]

Let $\tilde{u}_{n}(x)=u_{n}(x+y_{n})$ then there exists $\tilde{u},$ up to a subsequence, we have

[TABLE]

So, $\exists\Omega\subset B_{r}(0)$ $s.t.$ $\tilde{u}>0$ in $\Omega,$ we can infer

[TABLE]

Taking the limit as $n\rightarrow\infty$ and by applying Fatou’s lemma we obtain

[TABLE]

For any $\xi>0$, this gives a contradiction. Therefore, $\limsup\limits_{n\rightarrow\infty}t_{n}\leq 1.$ Case1:Assume that $\limsup\limits_{n\rightarrow\infty}t_{n}=1.$ Hence there exists a subsequence of $\{t_{n}\},$ still denoted by $\{t_{n}\}$ such that $t_{n}\rightarrow 1.$ Clearly,

[TABLE]

Moreover,

[TABLE]

Since $\{u_{n}\}$ is bounded in $H_{\varepsilon},$ by using the Mean Value Theorem and $t_{n}\rightarrow 1,$ we have

[TABLE]

For any $\xi$ and this gives a contradiction.

Case2: $\limsup\limits_{n\rightarrow\infty}t_{n}:=t_{0}<1.$ Then there exists a subsequence of $\{t_{n}\},$ still denoted by $\{t_{n}\}$ such that $t_{n}\rightarrow t_{0}$ and $t_{n}<1$ for any $n\in\mathbb{N},$ we deduce that

[TABLE]

This gives a contradiction.

By similar argument as the Lemma3.1 in [2] and Lemma4.7 in [45], we have the following lemma.

Lemma 3.9

Let $\{u_{n}\}$ be a sequence such that $u_{n}\rightharpoonup u$ in $H_{\varepsilon}$ and $w_{n}:=u_{n}-u.$ Then, we have

• $(i)$

$\int_{\mathbb{R}^{N}}|F(w_{n})-F(u_{n})+F(u)|^{t}dx=o_{n}(1)$* where $t=\frac{2N}{2N-\mu}.$*

• $(ii)$

$\int_{\mathbb{R}^{N}}(\frac{1}{|x|^{\mu}}\ast F(u_{n}-u))F(u_{n}-u)dx-\int_{\mathbb{R}^{N}}(\frac{1}{|x|^{\mu}}\ast F(u_{n}))F(u_{n})dx+\int_{\mathbb{R}^{N}}(\frac{1}{|x|^{\mu}}\ast F(u))F(u)dx=o_{n}(1).$**

• $(iii)$

$\forall$* $\xi>0,$ we have*

[TABLE]

• $(iv)$

[TABLE]

where, $\xi>0,$ $\varphi\in H_{\varepsilon}(\mathbb{R}^{N}).$

Proof:  $(i)$ By the Mean Value Theorem and (3.1), it follows that

[TABLE]

By applying Young inequality with $\delta>0,$ we get

[TABLE]

which yields

[TABLE]

[TABLE]

Let

[TABLE]

Then $G_{\delta,n}\rightarrow 0$ $a.e.$ in $\mathbb{R}^{N}$ as $n\rightarrow\infty$ and $0\leq G_{\delta,n}\leq C_{1}(|u|^{2t}+|u|^{qt})\in L^{1}(\mathbb{R}^{N}).$ As a consequence of the Dominated Convergence Theorem, we have

[TABLE]

On the other hand, from the definition of $G_{\delta,n}$, we get

[TABLE]

which together with the boundedness of $\{u_{n}\}$ gives

[TABLE]

As $\delta$ is arbitrary, we obtain

[TABLE]

$(ii)$

[TABLE]

By the boundedness of $\{u_{n}\}$ and $(f_{1})-(f_{2}).$ we know that

[TABLE]

From Lemma 2.3, we have

[TABLE]

Likewise, $I_{2}\rightarrow 0,$ $I_{3}\rightarrow 0.$ By the boundedness of $\{u_{n}\},$ we have $\{F(u_{n}-u)\}$ is bounded in $L^{\frac{2N}{2N-\mu}}(\mathbb{R}^{N})$ and $F(u_{n}-u)\rightarrow 0$ $a.e.$ in $\mathbb{R}^{N}.$ So, $F(u_{n}-u)\rightharpoonup 0$ in $L^{\frac{2N}{2N-\mu}}(\mathbb{R}^{N}).$ In view of \frac{1}{|x|^{\mu}}\ast F(u)\in\big{(}L^{\frac{2N}{2N-\mu}}(\mathbb{R}^{N})\big{)}^{\ast}, we obtain

[TABLE]

Therefore, we can conclude $(ii)$ holds.

$(iii)$ By using $(f_{1})$ and $(f_{2}),$ we know that for any $\xi>0,$ there exists $N_{0}\in(0,1)$ and $N_{1}>2$ such that

[TABLE]

[TABLE]

[TABLE]

Since $f$ is a continuous function, we deduce that exists $\delta\in(0,N_{0})$ such that

[TABLE]

Taking into account $u\in H_{\varepsilon}(\mathbb{R}^{N}),$ we know that there exists $R_{0}>0$ such that

[TABLE]

For any $\varphi\in H_{\varepsilon}(\mathbb{R}^{N}),$ $\|\varphi\|_{\varepsilon}=1,$ we have

[TABLE]

We denote A_{n}:=\big{\{}B^{c}_{R_{0}}(0):|u_{n}(x)|\leq N_{0}\big{\}}, B_{n}:=\big{\{}B^{c}_{R_{0}}(0):|u_{n}(x)|\geq N_{1}\big{\}}, C_{n}:=\big{\{}B^{c}_{R_{0}}(0):N_{0}<|u_{n}(x)|<N_{1}\big{\}}

[TABLE]

[TABLE]

[TABLE]

Thus, putting together (3.16), (3.17) and (3.18) we get

[TABLE]

Moreover,

[TABLE]

In view of $u\in H_{\varepsilon}$ we know that \big{|}(\mathbb{R}^{N}\setminus B_{R}(0))\bigcap\{|u|>\delta\}\big{|}\rightarrow 0 in $R\rightarrow\infty,$ then there exists $R_{1}>0$ $s.t.$ \big{|}(\mathbb{R}^{N}\setminus B_{R_{1}}(0))\bigcap\{|u|>\delta\}\big{|}<\xi. we define $R_{2}=\max\{R_{0},R_{1}\},$ we deduce that

[TABLE]

In similar way, we can prove that the follow inequality is true.

[TABLE]

Hence,

[TABLE]

It is easy to verify that

[TABLE]

Since $u_{n}$ is bounded, we have $u_{n}\rightarrow u$ $a.e.$ in $\mathbb{R}^{N},$ $u_{n}\rightarrow u$ in $L^{p}_{loc}(\mathbb{R}^{N}),$ $p\in[1,2^{*}_{s}).$ Let $l>q,$ such that $lt\in(2,2^{*}_{s}),$ $\frac{l}{l-1}t(q-1)\in(2,2^{*}_{s}).$ And

[TABLE]

Hence, we have

[TABLE]

Thus,

[TABLE]

Consequently,

[TABLE]

So, we obtain (3.19) hold. By applying Hölder inequality, for any $\xi>0,$ $n$ large enough, we have

[TABLE]

As a consequence, $\int_{\mathbb{R}^{N}}|f(u_{n}-u)-f(u_{n})+f(u)|^{t}|\varphi|^{t}dx\leq\xi\|\varphi\|^{t}_{\varepsilon}=\xi.$

$(iv)$

[TABLE]

Clearly, we have

[TABLE]

[TABLE]

In similar way, we get $|I_{3}|\leq C\xi\|\varphi\|_{\varepsilon}.$ Let us observe that,

[TABLE]

Since $\frac{1}{|x|^{\mu}}\ast F(u)\in L^{\frac{2N}{\mu}}(\mathbb{R}^{N}),$ we have $(\frac{1}{|x|^{\mu}}\ast F(u))^{2}\in L^{\frac{N}{\mu}}(\mathbb{R}^{N})$ and $(\frac{N}{\mu})^{\prime}=\frac{N}{N-\mu},$ $\frac{2N}{N-\mu}\in(2,2^{*}_{s}).$ so, we have $|u_{n}-u|^{2}\in L^{\frac{N}{N-\mu}}(\mathbb{R}^{N}).$ Since $|u_{n}-u|^{2}\rightharpoonup 0$ in $L^{\frac{N}{N-\mu}}(\mathbb{R}^{N}),$ we deduce that

[TABLE]

Moreover, we have $\frac{qt}{qt-1}\cdot\frac{qt-1}{qt}\cdot\frac{2N}{\mu}=\frac{2N}{\mu},$ $(\frac{2N(qt-1)}{qt\mu})^{\prime}=\frac{2N(qt-1)}{(2N-\mu)qt-2N},$ $\frac{(q-1)qt}{qt-1}\cdot\frac{2N(qt-1)}{(2N-\mu)qt-2N}=qt\in(2,2^{*}_{s})$ and $|u_{n}-u|^{\frac{(q-1)qt}{qt-1}}\rightharpoonup 0$ in $L^{\frac{2N(qt-1)}{(2N-\mu)qt-2N}}(\mathbb{R}^{N}),$ we get $(\frac{1}{|x|^{\mu}}\ast F(u))^{\frac{qt}{qt-1}}\in L^{\frac{2N(qt-1)}{qt\mu}}(\mathbb{R}^{N}),$ $|u_{n}-u|^{\frac{(q-1)qt}{qt-1}}\in L^{\frac{2N(qt-1)}{(2N-\mu)qt-2N}}(\mathbb{R}^{N})$ and $\int_{\mathbb{R}^{N}}(\frac{1}{|x|^{\mu}}\ast F(u))^{\frac{qt}{qt-1}}|u_{n}-u|^{\frac{(q-1)qt}{qt-1}}dx\rightarrow 0.$ Hence, we have

[TABLE]

In similar way, $|\int_{\mathbb{R}^{N}}(\frac{1}{|x|^{\mu}}\ast F(u_{n}-u))f(u)\varphi|\leq C\xi\|\varphi\|_{\varepsilon}.$ Therefore $(iv)$ holds.

By using Brezis-Lieb Lemma [10, 18] and Lemma 3.9, we have the following lemma.

Lemma 3.10

Let $\{u_{n}\}\subset H_{\varepsilon}$ be a $(PS)_{d}$ sequence of $J_{\varepsilon}$ with $u_{n}\rightharpoonup u$ in $H_{\varepsilon},$ then

• $(i)$

$J_{\varepsilon}(w_{n})=J_{\varepsilon}(u_{n})-J_{\varepsilon}(u)+o_{n}(1),$**

• $(ii)$

$\|J^{\prime}_{\varepsilon}(w_{n})\|=o_{n}(1).$**

Proof:  $(i)$ We note that

[TABLE]

By the Lemma3.9 $(ii)$, $u_{n}\rightharpoonup u$ in $H_{\varepsilon}$ and Brezis-Lieb Lemma. we have $(i)$ holds.

$(ii)$ Recall that $\{u_{n}\}$ is a $(PS)_{d}$ sequence of $J_{\varepsilon},$ we have $\|J^{\prime}_{\varepsilon}(u_{n})\|=o_{n}(1),$ $J^{\prime}_{\varepsilon}(u)=0.$ For any $\xi>0,$ $n$ large enough, $\forall$ $\varphi\in H_{\varepsilon}$ and $\|\varphi\|_{\varepsilon}=1,$ by the lemma3.9 $(iv)$ we get

[TABLE]

This completes the proof of $(ii)$

Lemma 3.11

$J_{\varepsilon}$* satisfies the $(PS)_{d}$ condition at any level $d\leq m_{{}_{V_{\infty}}}.$*

Proof:  Let $u_{n}\subset H_{\varepsilon}$ be a $(PS)_{d}$ sequence of $J_{\varepsilon}.$ Then, by Lemma 3.4 we know that $\{u_{n}\}$ is bounded in $H_{\varepsilon}$ and we can assume $u_{n}\geq 0.$ Hence, up to a subsequence, there is $u\in H_{\varepsilon}$ such that $u_{n}\rightharpoonup u\geq 0$ in $H_{\varepsilon},$ $u_{n}\rightarrow u$ in $L^{r}_{loc}(\mathbb{R}^{N})$ for each $r\in[2,2^{*}_{s}),$ $u_{n}(x)\rightarrow u(x)$ $a.e.$ in $\mathbb{R}^{N}$ and $J^{\prime}_{\varepsilon}(u)=0.$ Set $w_{n}=u_{n}-u,$ by Lemma 3.9 we have

[TABLE]

Moreover, for any $\alpha\in(2,2^{*}_{s})$ and $\alpha\leq 4,$ we have

[TABLE]

By Lemma 3.6 we have $d-J_{\varepsilon}(u)\leq d\leq m_{V_{\infty}}<\frac{s}{N}S^{\frac{N}{2s}}$ and by Lemma 3.8 we know $u_{n}\rightarrow u$ in $H_{\varepsilon}.$ Hence, the Lemma is proved.    

By Lemma 3.3 and Lemma 3.11 we have the following lemma.

Lemma 3.12

$J_{\varepsilon}|_{\mathcal{N}_{\varepsilon}}$* satisfies the $(PS)_{d}$ condition at any level $d<m_{V_{\infty}}.$*

$\mathbf{Proof\ of\ Theorem~{}\ref{Thm1.1}}.$ By Lemma 3.1 we know that functional $J_{\varepsilon}$ satisfies the mountain pass geometry, then using a version of the mountain pass theorem, there exists a sequence $\{u_{n}\}\subset H_{\varepsilon}$ such that

[TABLE]

For any $\tau\in\mathbb{R}$ with $V_{0}<\tau<V_{\infty},$ we have $m_{V_{0}}<m_{\tau}<m_{V_{\infty}}.$ By Lemma 3.6, $m_{\tau}<\frac{s}{N}S^{\frac{N}{2s}}.$ Apply Lemma 3.4, Lemma 3.11 and Theorem 6.3.4 in [47], we obtain that $m_{\tau}$ is a critical value of $I_{\tau}$ with corresponding nontrivial nonnegative critical point $u\in H_{\varepsilon}(\mathbb{R}^{N}).$ For any $r>0,$ take $\eta_{r}\in C^{\infty}_{0}(\mathbb{R}^{N},[0,1])$ be such that

[TABLE]

Set $u_{r}:=\eta_{r}u,$ it is easy to verify that $u_{r}\in H_{\varepsilon}(\mathbb{R}^{N})$ for each $r>0.$ By Lemma 3.2 there exists $t_{r}>0$ such that $\tilde{u}_{r}:=t_{r}u_{r}\in\mathcal{M}_{\tau}.$ Consequently, there is $r_{0}>0$ such that $\tilde{u}:=\tilde{u}_{r_{0}}$ satisfies $I_{\tau}(\tilde{u})<m_{V_{\infty}}.$ In fact, if this is false, then $I_{\tau}(\tilde{u}_{r})=I_{\tau}(t_{r}u_{r})\geq m_{V_{\infty}}$ for all $r>0.$ Notice that $u_{r}\rightarrow u$ in $H_{\varepsilon}(\mathbb{R}^{N})$ as $r\rightarrow+\infty$ and $u\in\mathcal{M}_{\tau}.$ we can deduce that $t_{r}\rightarrow 1$ as $r\rightarrow+\infty.$ Hence,

[TABLE]

which gives a contradiction, then $I_{\tau}(\tilde{u})<m_{{}_{V_{\infty}}}.$ The invariance by translation, we may assume $V_{0}=V(0)<\tau$ and $supp(\tilde{u})$ is compact. We use the continuity of $V,$ there is an $\varepsilon^{*}>0$ such that

[TABLE]

Hence,

[TABLE]

and

[TABLE]

Consequently,

[TABLE]

Lemma 3.11 guarantees up to a subsequence such that $u_{n}\rightarrow u$ in $H_{\varepsilon},$ then $J^{\prime}_{\varepsilon}(u)=0$ and $J_{\varepsilon}(u)=c_{\varepsilon}.$ Hence $u$ is a ground nontrivial nonnegative solution of (2.2). This completes the proof of Theorem 1.1.

4 Multiplicity Results

4.1 Technical results

In this section we focus our attention on the study of the multiplicity of solutions to (1.1). Since $V_{0}>0$, by Lemma 3.6, $m_{{}_{V_{0}}}<\frac{s}{N}S^{\frac{N}{2s}}.$ From the proof of Theorem 1.1 we know that $m_{{}_{V_{0}}}$ is a critical value of $I_{V_{0}}$ with corresponding nontrivial nonnegative critical point $w\in H^{s}(\mathbb{R}^{N}).$ Fix $\delta>0$ and let $\eta\in C^{\infty}(\mathbb{R}^{+},[0,1])$ be a function such that $\eta(t)=1$ if $0\leq t\leq\frac{\delta}{2}$ and $\eta(t)=0$ if $t\geq\delta.$ For any $y\in\Lambda,$ we define

[TABLE]

Then for small $\varepsilon>0$, one has $\Psi_{\varepsilon,y}\in H_{\varepsilon}\backslash\{0\}$ for all $y\in\Lambda$. In fact, using the change of variable $z=x-\frac{y}{\varepsilon}$, one has

[TABLE]

Moreover, using the change of variable $x^{\prime}=x-\frac{y}{\varepsilon},z^{\prime}=z-\frac{y}{\varepsilon}$, we have

[TABLE]

where $\eta_{\varepsilon}(x)=\eta(|\varepsilon x|)$. By Lemma 2.4, we see that $\eta_{\varepsilon}w\in\mathcal{D}^{s,2}(\mathbb{R}^{N})$ as $\varepsilon\rightarrow 0$, and hence $\Psi_{\varepsilon,y}\in\mathcal{D}^{s,2}(\mathbb{R}^{N})$ for $\varepsilon>0$ small. Hence $\Psi_{\varepsilon,y}\in H_{\varepsilon}$. Now we proof $\Psi_{\varepsilon,y}\neq 0$. In fact,

[TABLE]

as $\varepsilon\rightarrow 0$. Then $\Psi_{\varepsilon,y}\neq 0$ for small $\varepsilon>0$. Therefore, there exists unique $t_{\varepsilon}>0$ such that

[TABLE]

We introduce the map $\Phi_{\varepsilon}:\Lambda\rightarrow\mathcal{N}_{\varepsilon}$ by setting

[TABLE]

By construction, $\Phi_{\varepsilon}(y)$ has a compact support for any $y\in\Lambda$ and $\Phi_{\varepsilon}$ is a continuous map.

Lemma 4.1

[TABLE]

Proof:  Assume by contradiction, then there exists $\delta_{0}>0,$ $\{y_{n}\}\subset\Lambda$ and $\varepsilon_{n}>0$ with $\varepsilon_{n}\rightarrow 0$ such that

[TABLE]

By using $\Phi_{\varepsilon_{n}}\in\mathcal{N}_{\varepsilon_{n}}$ and Lemma 3.5 we know that there is a $r_{0}>0$ such that

[TABLE]

which implies that $t_{\varepsilon}\nrightarrow 0.$ Hence there exists a $T>0$ such that $t_{\varepsilon_{n}}\geq T.$ If $t_{\varepsilon_{n}}\rightarrow\infty,$ we have

[TABLE]

for large $n.$ This yield a contradiction, then $t_{\varepsilon}\rightarrow t_{0}>0.$ Now we claim that $t_{0}\rightarrow 1.$ By using Lebesgue’s theorem, we can verify that

[TABLE]

[TABLE]

and

[TABLE]

Therefore, from (4.2), we get

[TABLE]

This show $t_{0}w\in\mathcal{M}_{V_{0}}.$ Noting that $w\in\mathcal{M}_{V_{0}},$ we see $t_{0}=1,$ so claim is proved. Moreover, similar to the above arguments, we can get

[TABLE]

which contradicts to (4.1). This completes the proof.

Now, we are ready to introduce the barycenter map. For any $\delta>0,$ let $\rho=\rho(\delta)>0$ such that $\Lambda_{\delta}\subset B_{\rho}(0).$ Define $\Upsilon:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ as follow:

[TABLE]

We define the barycenter map $\beta_{\varepsilon}:\mathcal{N}_{\varepsilon}\rightarrow\mathbb{R}^{N}$ as follows

[TABLE]

Lemma 4.2

[TABLE]

Proof:  Assume by contradiction, then there exists $\delta_{0}>0,$ $\{y_{n}\}\subset\Lambda$ and $\varepsilon_{n}\rightarrow 0^{+}$ such that

[TABLE]

By using the definitions of $\beta_{\varepsilon_{n}}$ and $\Phi_{\varepsilon_{n}},$ we can see that

[TABLE]

Taking into account the Lebesgue dominant convergence theorem, we can infer that

[TABLE]

which contradicts (4.3).

Lemma 4.3

For any $\tau>0,$ let $\{u_{n}\}\subset\mathcal{M}_{\tau}$ with $I_{\tau}(u_{n})\rightarrow m_{\tau}.$ Then $\{u_{n}\}$ has a subsequence strongly convergent in $H^{s}(\mathbb{R}^{N}).$ Particulary, there exists a minimizer for $m_{\tau}.$

Proof:  From the proof of Lemma 3.4 and Lemma 3.6, we know that $\{u_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N})$ and $m_{\tau}<\frac{s}{N}S^{\frac{N}{2s}}.$ By the Ekeland Variational principle, we may assume that $\{u_{n}\}$ is a $(PS)_{m_{\tau}}$ sequence of $I_{\tau}.$ Then, by Lemma 3.8, there exists $u\in H^{s}(\mathbb{R}^{N})$ such that, up to a subsequence, $u_{n}\rightarrow u$ in $H^{s}(\mathbb{R}^{N}).$ Moreover, $u$ is a minimizer of $m_{\tau}.$

Lemma 4.4

Let $\varepsilon_{n}\rightarrow 0$ and $u_{n}\in\mathcal{N}_{\varepsilon_{n}}$ be such that $J_{\varepsilon_{n}}(u_{n})\rightarrow m_{{}_{V_{0}}}.$ Then there exists a sequence $\{y_{n}\}\subset\mathbb{R}^{N}$ such that $u_{n}(\cdot+y_{n})$ has a convergent subsequence in $H^{s}(\mathbb{R}^{N}).$ Moreover, up to a subsequence, $\tilde{y}_{n}=\varepsilon_{n}y_{n}\rightarrow y\in\Lambda.$

Proof:  Since $u_{n}\in\mathcal{N}_{\varepsilon_{n}}$ and $\lim\limits_{n\rightarrow\infty}J_{\varepsilon_{n}}(u_{n})=m_{{}_{V_{0}}},$ by Lemma 3.4 we can see that $\{u_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N}).$ By Lemma 3.5, we have $\|u_{n}\|_{\varepsilon_{n}}\nrightarrow 0.$ we can argue as in Lemma 3.7 to obtain a sequence $\{y_{n}\}$ and constant $r>0$ such that

[TABLE]

Note, if this is false, then for any $r>0,$ we have

[TABLE]

By Lemma 2.2, we know that $u_{n}\rightarrow 0$ in $L^{t}(\mathbb{R}^{N})$ for $t\in[2,2^{*}_{s}),$ we can argue as the proof of (3.2) and we deduce that

[TABLE]

As the proof of Lemma 3.7, we can prove $\int_{\mathbb{R}^{N}}|u|^{2^{*}_{s}}dx=o_{n}(1).$ Since $u_{n}\in\mathcal{N}_{\varepsilon_{n}},$ we get $\|u_{n}\|_{\varepsilon_{n}}=o_{n}(1),$ which gives a contradiction. Hence, (4.4) holds. Now, we set $\tilde{u}_{n}=u_{n}(\cdot+y_{n}).$ Since, $\{u_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N})$ and (4.4), up to a subsequence, we have $\tilde{u}_{n}\rightharpoonup\tilde{u}\neq 0$ in $H^{s}(\mathbb{R}^{N})$ and $\tilde{u}_{n}(x)\rightarrow\tilde{u}(x)$ $a.e.$ in $\mathbb{R}^{N}.$ Fix $t_{n}>0$ such that $t_{n}\tilde{u}_{n}\in\mathcal{M}_{V_{0}}$ and set $\tilde{y}_{n}=\varepsilon_{n}y_{n}.$ Since $u_{n}\in\mathcal{N}_{\varepsilon_{n}},$ we can see that

[TABLE]

which gives

[TABLE]

By Lemma 4.3, up to subsequence, we get $t_{n}\tilde{u}_{n}:=v_{n}\rightarrow v_{0}$ in $H^{s}(\mathbb{R}^{N}).$ Note,

[TABLE]

For large $n,$ we have $0<\frac{\beta}{2}<\|\tilde{u}_{n}\|^{2}_{H^{s}(\mathbb{R}^{N})},$ then

[TABLE]

Hence $\{t_{n}\}$ is bounded, and we may assume that $t_{n}\rightarrow t^{*}>0.$ So, up to a subsequence, we have

[TABLE]

In order to complete the proof of the lemma, we show that $\{\tilde{y}_{n}\}$ is bounded in $\mathbb{R}^{N}.$ We argue by contradiction, up to a subsequence, we assume that $|\tilde{y}_{n}|\rightarrow\infty.$ Notice that, up to subsequence, we have $v_{n}\rightarrow v_{0}\neq 0$ in $H^{s}(\mathbb{R}^{N}).$ By Fatou’s lemma we get

[TABLE]

which is a contradiction, so we get $\{\tilde{y}_{n}\}$ is bounded in $\mathbb{R}^{N}.$ Therefore, up to subsequence, $\tilde{y}_{n}\rightarrow y\in\mathbb{R}^{N}.$ If $y\in\mathbb{R}^{N}\setminus\Lambda$ then $V_{0}<V(y).$ This is a contradiction. Hence, we can conclude that $y\in\Lambda.$

Now, we introduce a subset $\tilde{\mathcal{N}}_{\varepsilon}$ of $\mathcal{N}_{\varepsilon}$ by setting

[TABLE]

where $h(\varepsilon):=\max\limits_{y\in\Lambda}|J_{\varepsilon}(\Phi_{\varepsilon}(y))-m_{{}_{V_{0}}}|.$ Then, we can use Lemma 4.1 to conclude that

[TABLE]

Hence, for each $y\in\Lambda$ and $\varepsilon>0,$ we have $\Phi_{\varepsilon}(y)\in\tilde{\mathcal{N}}_{\varepsilon}.$ By Lemma 4.4, we can prove the following Lemma.

Lemma 4.5

For any $\delta>0,$ we have

[TABLE]

Proof:  Let $\varepsilon_{n}\rightarrow 0.$ For any $n\in\mathbb{N},$ there exists $\{u_{n}\}\subset\mathcal{\tilde{N}}_{\varepsilon_{n}}$ such that

[TABLE]

Since $\{u_{n}\}\in\mathcal{N}_{\varepsilon_{n}},$ it follow that

[TABLE]

Then, $J_{\varepsilon_{n}}(u_{n})\rightarrow m_{{}_{V_{0}}}.$ By Lemma 4.4, there exists $\{y_{n}\}\in\mathbb{R}^{N}$ such that $\{\tilde{u}_{n}(\cdot):=u_{n}(\cdot+y_{n})\}$ has a convergent subsequence in $H^{s}(\mathbb{R}^{N})$ and $\tilde{y}_{n}:=\varepsilon_{n}y_{n}\rightarrow y\in\Lambda.$ Then,

[TABLE]

The proof is completed.

4.2 Proof of Theorem1.2

Lemma 4.6

Assume that $(V)$ and $(f_{1})-(f_{4})$ hold. Then, for any $\delta>0$ there exists $\varepsilon_{\delta}>0$ such that the problem (1.1) has at least $cat_{\Lambda_{\delta}}(\Lambda)$ nontrivial nonnegative solutions for all $\varepsilon\in(0,\varepsilon_{\delta}).$

Proof:  By Lemma 4.1 and the define of $\psi_{\varepsilon},$ we have

[TABLE]

Then, there exists $\varepsilon_{1}>0$ such that $\tilde{\mathcal{S}}_{\varepsilon}:=\{u\in\mathcal{S}_{\varepsilon}:\psi_{\varepsilon}(u)\leq m_{{}_{V_{0}}}+h(\varepsilon)\}\neq 0$ for all $\varepsilon\in(0,\varepsilon_{1}).$

Applying Lemma 4.1, Lemma 3.3, Lemma 4.2 and Lemma 4.5, we can find some $\varepsilon_{1}=\varepsilon_{\delta}>0$ such that the following diagram

[TABLE]

is well defined for any $\varepsilon\in(0,\varepsilon_{1}).$By the proof of [7, Theorem5.1,Theorem5.2], we know that for $\varepsilon>0$ small enough, we deduce from Lemma 3.12 that $\psi_{\varepsilon}$ satisfies the $PS$ condition in $\tilde{\mathcal{S}}_{\varepsilon}.$ And $\psi_{\varepsilon}$ has at least $cat_{\tilde{\mathcal{S}}_{\varepsilon}}(\tilde{\mathcal{S}}_{\varepsilon})$ critical points on $\tilde{\mathcal{S}}_{\varepsilon}.$ By Lemma 3.3 we conclude that $J_{\varepsilon}$ admits at least $cat_{\Lambda_{\delta}}(\Lambda)$ critical points on $\mathcal{N}_{\varepsilon}$.

Now, we use a Moser iteration argument [31] to study of behavior of the maximum points of the solutions.

Lemma 4.7

Let $\varepsilon_{n}\rightarrow 0$ and $u_{n}\in\tilde{\mathcal{N}}_{\varepsilon_{n}}$ is a nontrivial nonnegative solution to (2.2). Then exists $y_{n}\in\mathbb{R}^{N}$ such that $v_{n}=u_{n}(\cdot+y_{n})$ satisfies the following problem

[TABLE]

where $V_{n}(x)=V(\varepsilon_{n}x+\varepsilon_{n}y_{n}),$ $\varepsilon_{n}y_{n}\rightarrow y\in\Lambda$ and there exists $C>0$ such that $\|v_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq C$ for all $n\in\mathbb{N}.$ Furthermore,

[TABLE]

Proof:  For any $L>0$ and $\beta>1,$ let us define the function

[TABLE]

where $v_{L,n}=\min\{v_{n},L\}.$ Since $r$ is an increasing function in $(0,+\infty),$ then we have

[TABLE]

Define the functions

[TABLE]

For all $a,b\in\mathbb{R}$ such that $a>b,$ by applying Jensen inequality we get

[TABLE]

In similar way, we can prove that the above inequality is true for all $a\leq b.$ Therefore

[TABLE]

By using (4.6), we have

[TABLE]

Now, we take $r(v_{n})=v_{n}v_{L,n}^{2(\beta-1)}$ as test-function in (4.5) and in view of (4.7), we obtain

[TABLE]

Since

[TABLE]

and we can use Lemma 2.1 to deduce that

[TABLE]

On the other hand, since $\{v_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N}),$ there exists $C_{0}>0$ such that

[TABLE]

Taking $\xi\in(0,V_{0}),$ and using (3.1), (4.9) and (4.10), we can see that (4.8) yields

[TABLE]

Set $q+2\beta-2=2^{*}_{s}$ $\Rightarrow$ $\beta=\frac{1}{2}(2^{*}_{s}+2-q)>1,$ then

[TABLE]

By $\{u_{n}\}$ is bounded in $H_{s}$, $\exists R_{0}>0$ $s.t.$ \big{(}\int_{\{v_{n}>R_{0}\}}|v_{n}|^{2^{*}_{s}}dx\big{)}^{\frac{2^{*}_{s}-2}{2^{*}_{s}}}\leq\frac{1}{2C\beta^{2}}. Hence, we can see that

[TABLE]

Therefore, we can deduce that

[TABLE]

Taking the limit in (4.11) as $L\rightarrow+\infty$ and Fatou lemma, we have \big{(}\int_{\mathbb{R}^{N}}|v_{n}|^{2^{*}_{s}\beta}dx\big{)}^{\frac{2}{2^{*}_{s}}}\leq C<+\infty. so, $v_{n}\in L^{2^{*}_{s}\beta}(\mathbb{R}^{N}).$ For any $\beta>\frac{1}{2}(2^{*}_{s}+2-q)>1$ and $\beta\leq 1+\frac{2^{*}_{s}}{2}\cdot\frac{2^{*}_{s}-q}{2}$ then $2<q+2\beta-2<2^{*}_{s}+2\beta-2\leq 2^{*}_{s}(1+\frac{2^{*}_{s}-q}{2}).$ we can deduce that

[TABLE]

Let $a=\frac{2^{*}_{s}(2^{*}_{s}-q)}{2(\beta-1)},$ $b=q+2\beta-2-a,$ $r=\frac{2^{*}_{s}}{a},$ $r^{\prime}=\frac{2^{*}_{s}}{2^{*}_{s}-a},$ then $\frac{2^{*}_{s}b}{2^{*}_{s}-a}=2^{*}_{s}+2\beta-2.$ Taking into account Young inequality we have

[TABLE]

Therefore,

[TABLE]

We note to $\beta>1,$ we deduce that

[TABLE]

Now, we set $\beta=1+\frac{2^{*}_{s}}{2}\cdot\frac{2^{*}_{s}-q}{2},$ then observing that $2^{*}_{s}+2\beta-2=2^{*}_{s}(\frac{1}{2}(2^{*}_{s}+2-q)).$ Iterating this process and recalling that $2^{*}_{s}+2\beta_{i-1}-2=2^{*}_{s}\beta_{i}.$ Argue as [21]. Thus,

[TABLE]

Replacing it in (4.12) we have

[TABLE]

Denoting $C_{i+1}=C\beta_{i+1}^{2}$ and $K_{i}:=(1+\int_{\mathbb{R}^{N}}v_{n}^{2\beta_{i}+2^{*}_{s}-2}dx)^{\frac{1}{2(\beta_{i}-1)}}.$ We conclude that there exists a constant $C_{0}>0$ independent of $i,$ such that

[TABLE]

Therefore,

[TABLE]

uniformly on $n\in\mathbb{N},$ thanks to $v_{n}\in L^{2^{*}_{s}\beta_{1}}(\mathbb{R}^{N})$ and $\|v_{n}\|_{\varepsilon_{n}}\leq C.$ Arguing as in [4], we can prove that

[TABLE]

Now we consider $\varepsilon_{n}\rightarrow 0^{+}$ and take a sequence $u_{n}\in\tilde{\mathcal{N}}_{\varepsilon}$ of solutions of the problem (2.2) as above. There exists $\gamma>0$ such that

[TABLE]

Assume by contradiction, we have $\lim\limits_{n\rightarrow\infty}\|u_{n}\|_{L^{\infty}(\mathbb{R}^{N})}=0.$ For any $\xi>0,$ there exists $n_{0}$ such that $\|u_{n}\|_{L^{\infty}(\mathbb{R}^{N})}<\xi$ for any $n>n_{0}.$ Since $u_{n}\in\tilde{\mathcal{N}}_{\varepsilon},$ we have

[TABLE]

where $t=\frac{2N}{2N-\mu}.$ Since $2t\in(2,2^{*}_{s})$ and $qt\in(2,2^{*}_{s}),$ there exists $\sigma>0$ small enough such that $(2t-\sigma)\in(2,2^{*}_{s})$ and $(qt-\sigma)\in(2,2^{*}_{s}).$ Since we have that $\{u_{n}\}$ is bound in $H_{0}^{1}(\mathbb{R}^{N}),$ we can deduce to

[TABLE]

This implies that $\|u_{n}\|_{\varepsilon_{n}}\rightarrow 0~{}(n\rightarrow\infty).$ In similar way, we can decude

[TABLE]

then $J_{\varepsilon_{n}}(u_{n})\rightarrow 0~{}(n\rightarrow\infty),$ this contradict with $J_{\varepsilon_{n}}(u_{n})\rightarrow m_{{}_{V_{0}}}>0.$ As a consequence, (4.13) holds. By Lemma 4.7, we have

[TABLE]

and

[TABLE]

There exists $R>0$ such that $\|v_{n}\|_{L^{\infty}(B^{c}_{R}(0))}<\gamma,$ then

[TABLE]

Hence

[TABLE]

Let $p_{n}$ is the global maximum point of $u_{n},$ taking into account (4.14) and (4.15) we can get $p_{n}\in B_{R}(y_{n}).$ Hence, $p_{n}=y_{n}+q_{n}$ for some $q_{n}\in B_{R}(0).$ Then $\xi_{\varepsilon_{n}}=\varepsilon_{n}y_{n}+\varepsilon_{n}q_{n}$ is the maximum point of $u_{n}(\frac{x}{\varepsilon_{n}}).$ Since $|q_{n}|<R$ for any $n\in\mathbb{N}$ and $\varepsilon_{n}y_{n}\rightarrow y_{0}\in\Lambda.$ Therefore,

[TABLE]

which ends the proof of the Theorem1.2.

Acknowledgment

We would like to thank the anonymous referee for his/her careful readings of our manuscript and the useful comments made for its improvement.