Existence and concentration of positive solutions for a p-fractional Choquard equation

1.   Introduction

In this paper, we consider the following nonlinear equation governed by the fractional $p$-Laplacian

where $\varepsilon > 0$ is a small parameter, $N > ps$, $s \in (0, 1)$, $1 < p < \infty$, $p < q < \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N})$ and $V, K$ are positive functions. $(-\Delta)^{s}_{p}$ denotes the fractional $p$-Laplacian defined for all $u:\mathbb{R}^{N} \rightarrow \mathbb{R}$ smooth enough by

the $P.V.$ stands for the Cauchy principle value (see ^[10]).

In the case $s = 1$, $p = 2$ and $K(x)\equiv1$, the Eq (1.1) boils down to the Choquard equation

When $N = 3$, $\mu = 1$ and $q = 2$, the Eq (1.2) has appeared in the several context of quantum physics, such as the description of a Polaron at rest ^[20] and the model of an electron trapped in its own hole ^[13] and the coupling of the Schrödinger equation under a classical Newtonian gravitational potential ^[12]. The pioneering mathematical research goes back to Lieb ^[13] and Lions ^[14]. The existence and qualitative of solutions of equation like (1.2) have been extensively studied by variational methods, see for example ^{[16,17,18,19,29]} and their references. For the existence of semi-classical solutions to Choquard equation (1.2) were studied in some papers. In ^[28], Wei and Winter constructed a family of solutions which concentrate to the non-degenerate critical points of the potential $V$. Moroz and Schaftingen ^[18] proved the existence of solutions concentrating around the local minimum of $V$ by a nonlocal penalization method. See ^[3] for the existence and multiplicity for a generalized quasilinear Choquard equation.

In recent years, a great attention has been given to problems driven by the fractional Laplacian. One of the reasons for this comes from the fact that this operator appears in several applications in different subjects, such as crystal dislocation, thin obstacle problems, optimization and finance, anomalous diffusion and many others, we can see ^[10,25]. Recently, d'Avenia, Siciliano and Squassina ^[21] considered the existence, regularity, symmetry as well as decay properties of the following fractional Choquard equation

Shen, Gao and Yang ^[23] obtained the existence of ground states of (1.3) with the nonlinearity satisfies the generaal Beresty-Lions type assumptions. Zhang and Wu ^[31] studied the existence of nodal solutions of (1.3). Chen and Liu ^[5] studied (1.3) with nonconstant linear potential and proved the existence of ground states without any symmetry property. Ambrosio ^[1] investigated the multiplicity and concentration of positive solutions for a fractional Choquard equation with general nonlinearity. In ^[7], Chen, Li and Yang obtained the multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent. For other existence results we refer to ^[4,11,24,32] and the references therein.

To the best of our knowledge, there are few results about fractional Choquard equation like (1.3). Belchior et al. ^[8] investigated the following equation

where $F$ is the primitive of $f$ and $A$ is a positive constant. They showed the existence of ground states and asymptotic of the solutions for (1.4). In ^[2], Ambrosio studied the following problem

He proved solutions concentrating around global minimum of the potential $V$.

Recently, Wang et al. ^[27] applied a kind of structure introduced by Ding and Liu ^[9] to study the existence and concentration of positive solutions for semilinear Schrödinger-Poisson system. The similar results for the fractional Schrödinger-Poisson system, we can see ^[30]. Alves and Yang ^[3] considered the generalized quasilinear Choquard equation

where $1 < p < N$, $V$ and $Q$ are two continuous functions satisfy the structure of ^[9], they established concentration behavior for the Choquard equation. It is quite natural to ask how the potentials will affect the existence and concentration of solutions for (1.1). In this paper, we shall give an affirmative answers for this question.

Motivated by the above papers, we will establish the existence, multiplicity and concentration of positive solutions for Eq (1.1). To gain further insight into the effect of potential functions $V$ and $K$ on the concentration process, we give the following assumptions introduced by ^[9]. Set

We assume that $V$ and $K$ satisfy:

($\text{H}_{1}$) $V, K\in L^{\infty}(\mathbb{R}^{N})$ are uniformly continuous and $\theta > 0$, $\inf_{x\in \mathbb{R}^{N}}K(x) > 0$.

($\text{H}_{2}$) $\theta < V_{\infty} < \infty$ and there exist $R > 0, x_{*}\in \mathcal{V}$ such that

($\text{H}_{3}$) $\kappa > K_{\infty} \geq \inf_{x\in \mathbb{R}^{N}}K(x)$ and there exist $R > 0, x^{*}\in \mathcal{K}$ such that

From ($\text{H}_{2}$), we may assume that $K(x_{*}) = \max_{x\in \mathcal{V}}K(x)$. Set

From ($\text{H}_{3}$), we may assume that $V(x^{*}) = \min_{x\in \mathcal{K}}V(x)$. Set

Clearly, $\Omega_{V}$ and $\Omega_{K}$ are bounded sets. Moreover, if $\mathcal{V}\cap \mathcal{K} \neq \emptyset$, then $\Omega_{V} = \Omega_{K} = \mathcal{V}\cap \mathcal{K}$. In particular, $\Omega_{V} = \mathcal{V}$ if $K(x)$ is a constant, and $\Omega_{K} = \mathcal{K}$ if $V(x)$ is a constant.

We now state our main results.

Theorem 1.1. Assume that ($\mathit{\text{H}}_{1}$) and ($\mathit{\text{H}}_{2}$) hold, then for all small $\varepsilon > 0$, (1.1) has a positive ground state solution $u_{\varepsilon}$, and there exists a maximum point $x_{\varepsilon}$, such that up to a subsequence, $x_{\varepsilon} \rightarrow x_{0}$ as $\varepsilon \rightarrow 0$, $\lim_{\varepsilon \rightarrow 0}dist(x_{\varepsilon}, \Omega_{V}) = 0$, and $v_{\varepsilon}(x) = u_{\varepsilon}(\varepsilon x + x_{\varepsilon})$ converges in $W^{s, p}(\mathbb{R}^{N})$ to a ground state solution of

In particular, if $\mathcal{V}\cap \mathcal{K} \neq \emptyset$, then $\lim_{\varepsilon \rightarrow 0}dist(x_{\varepsilon}, \mathcal{V}\cap \mathcal{K}) = 0$, and up to a subsequence, $v_{\varepsilon}$ converges in $W^{s, p}(\mathbb{R}^{N})$ to a ground state solution of

If ($\mathit{\text{H}}_{1}$) and ($\mathit{\text{H}}_{3}$) hold, and we replace $\Omega_{V}$ by $\Omega_{K}$, then all the conclusions remain true.

Let $\mathcal{V}\cap \mathcal{K} \neq \emptyset$. Now we denote $\Lambda = \mathcal{V}\cap \mathcal{K}$. It is easy to check that $\Lambda$ is compact. For any $\delta > 0$, set $\Lambda_{\delta} = \{x\in \mathbb{R}^{N}: dist(x, \Lambda)\leq \delta\}$.

Theorem 1.2. Assume that ($\mathit{\text{H}}_{1}$) and ($\mathit{\text{H}}_{2}$) or ($\mathit{\text{H}}_{3}$) hold, then for all small $\varepsilon > 0$, problem (1.1) has at least $cat_{\Lambda_{\delta}}(\Lambda)$ solutions, if $x_{\varepsilon}$ its global maximum, up to a subsequence, such that $\lim_{\varepsilon \rightarrow 0}dist(x_{\varepsilon}, \Lambda) = 0,$ $u_{\varepsilon}$ converges in $W^{s, p}(\mathbb{R}^{N})$ to a ground state solution of

Note that our main results are also new for the case $p = 2$. Our main theorem improves the result in ^[2,8] with both linear potential $V$ and nonlinear potential $K$ of the concentration behavior of positive solutions. There are some difficulties in such a problem. The first one is that there would presumably be competition between the $V$ and $K$: each would try to attract ground states to their minimum and maximum points, respectively. The second one, the operator $(-\Delta)^{s}_{p}$ and the convolution term are all nonlocal operators, make our analysis more complicated with respect to ^[3], so we need more accurate estimates.

The plan of this paper is the following: In Section 2, we give some preliminary results which will be used later. In Section 3, we show some compactness lemmas of the functional associated to our problem. In Section 4, we consider the existence of ground states of case of (1.1) and the concentration phenomenon. In the final section, we prove Theorem 1.2.

In this paper, we will use the following notations:

The notations $C$, $C_{1}, C_{2}\cdot\cdot\cdot$ are positive (possibly different) constants.

$B_{r}(z_{0})$ denotes the ball in $\mathbb{R}^{N}$ centered at $z_{0}$ with radius $r$.

$o_{n}(1)$ and $o_{\varepsilon}(1)$ denotes the vanishing quantities as $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$.

We will use $\|\cdot\|_{q}$ for the norm in $L^{q}(\mathbb{R}^{N})$, $u^{+} = \max\{u, 0\}$ and $u^{-} = \min\{u, 0\}$.

2.   Preliminaries

In this section, we recall some known results for the readers convenience and the later use. First, we will give some useful facts for the fractional order Sobolev spaces. Let $0 < s < 1 < p < \infty$ be real numbers, the homogeneous fractional Sobolev space $\mathcal{D}^{s, p}(\mathbb{R}^{N})$ as the completion of $C^{\infty}_{0}(\mathbb{R}^{N})$ with respect to the Gagliardo seminorm

The fractional Sobolev space $W^{s, p}(\mathbb{R}^{N})$ is defined as

equipped with the norm

It is easy to see that the embedding $W^{s, p}(\mathbb{R}^{N}) \hookrightarrow L^{r}(\mathbb{R}^{N})$ is continuous for any $r\in [p, p^{*}_{s}]$, and compactly in $L_{loc}^{r}(\mathbb{R}^{N})$ for any $r\in [p, p^{*}_{s})$.

Making the change of variable $x\mapsto \varepsilon x$, Eq (1.1) becomes

where $V_{\varepsilon}(x) = V(\varepsilon x)$ and $K_{\varepsilon}(x) = K(\varepsilon x)$. Eqs (1.1) and (2.1) are equivalent, we shall thereafter focus on Eq (2.1). For any $\varepsilon > 0$, let $E_{\varepsilon} = \{u\in W^{s, p}(\mathbb{R}^{N}): \int_{\mathbb{R}^{N}}V_{\varepsilon}(x)|u|^{p}dx < \infty \}$ be the Sobolev space endowed with the norm

By the assumption of $V$, we see that $\|\cdot\|_{\varepsilon}$ and $\|\cdot\|$ are equivalent norms for $\varepsilon > 0$. Define the energy functional associated with (2.1) by

Note that $p < q < \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N})$, by the Hardy-Littlewood-Sobolev inequality (^[15]) and the boundedness of $K$, we have

Therefore, the functional $I_{\varepsilon}$ is well defined on $E_{\varepsilon}$ and belongs to $\mathcal{C}^{1}(E_{\varepsilon}, \mathbb{R})$.

Define the solution manifold of (2.1) by

For any $u\in \mathcal{N}_{\varepsilon}$, by (2.2) we have

for some $r^{*} > 0$.

The ground energy associated with (2.1) is defined as

The following vanishing lemma is a version of the Concentration-compactness principle of P. L. Lions. We can see (Lemma 2.1 of ^[2]).

Lemma 2.1. Let $N > ps$. Assume that $\{u_{n}\}$ is bounded in $W^{s, p}(\mathbb{R}^{N})$ and it satisfies

for some $R > 0$. Then $u_{n} \rightarrow 0$ strongly in $L^{r}(\mathbb{R}^{N})$ for every $r\in (p, p_{s}^{*})$.

From (2.2) and $q > p$, it follows that $I_{\varepsilon}$ satisfies the geometry of the mountain pass (see ^[26]). Hence, there is a sequence $\{u_{n}\}\subset E_{\varepsilon}$ such that

where $c^{*}_{\varepsilon}$ is the mountain pass level given by

and $\Gamma = \{\gamma\in C^{1}([0, 1], E_{\varepsilon}): \gamma(0) = 0, I_{\varepsilon}(\gamma(1)) < 0\}$.

We observe that for any $u \in E_{\varepsilon}\setminus\{0\}$, there exists a unique $t_{u} > 0$ such that $t_{u}u\in \mathcal{N}_{\varepsilon}$, and the maximum of the function $g(t) = I_{\varepsilon}(tu)$ for $t\geq 0$ is achieved at $t = t_{u}$. By a standard arguments, we have

For any $a, b > 0$, consider the limit problem

Solutions of (2.5) are critical points of the functional defined by

Define the solution manifold of (2.5) by

The ground energy associated with (2.5) is defined as $c_{ab} = \inf_{u\in \mathcal{M}_{ab}}I_{ab}(u)$. It is easy to check that

By ^[8], we known that (2.5) has a positive ground state solution $\omega$, that is $c_{ab} = I_{ab}(\omega)$.

Lemma 2.2. Let $a_{1}, a_{2} > 0$ and $b_{1}, b_{2} > 0$, with $a_{1} \leq a_{2}$ and $b_{1}\geq b_{2}$. Then $c_{a_{1}b_{1}} \leq c_{a_{2}b_{2}}$. In particular, if one of inequalities is strict, then $c_{a_{1}b_{1}} < c_{a_{2}b_{2}}$.

Proof. Let $u \in \mathcal{M}_{a_{2}b_{2}}$ be a ground state solution of (2.5) with coefficients $a_{2}, b_{2}$ such that

It is easy to check that there exists $t_{0} > 0$ such that $t_{0}u\in \mathcal{M}_{a_{1}b_{1}}$. Then we get

It follows from (2.6) and (2.7) that

The proof is completed.

3.   A compactness condition

In this section we will show some compactness results for the functional $I_{\varepsilon}$.

Lemma 3.1. $\{u_{n}\} \subset E_{\varepsilon}$ is a $(PS)_{c}$ sequence for $I_{\varepsilon}$ with $u_{n}\rightharpoonup 0$ weakly in $E_{\varepsilon}$. If $u_{n}\not\rightarrow 0$ in $E_{\varepsilon}$, the $c\geq c_{\infty}: = c_{V_{\infty}K_{\infty}}$.

Proof. Let $\{u_{n}\}$ be a $(PS)_{c}$ sequence for $I_{\varepsilon}$, by (2.4), we have

for $n$ large enough. Therefore $\{u_{n}\}$ is bounded in $E_{\varepsilon}$.

For each $n$, there is a unique $t_{n} > 0$ such that $t_{n}u_{n}\in \mathcal{M}_{V_{\infty}K_{\infty}}$. We now show that the sequence $\{t_{n}\}$ satisfies $\limsup_{n\rightarrow \infty} t_{n} \leq 1$. By contradiction we assume that there exist $\sigma > 0$ and a subsequence (still denoted by $\{t_{n}\}$) such that $t_{n} \geq 1+ \sigma$ for all $n$. From the boundedness of $\{u_{n}\}$, we have $\langle I'_{\varepsilon}(u_{n}), u_{n}\rangle = o_{n}(1)$. That is

Since $t_{n}u_{n}\in \mathcal{M}_{V_{\infty}K_{\infty}}$, we obtain

We deduce from (3.2) and (3.3) that

By the definition of $V_{\infty}$ and $K_{\infty}$, for any $\nu > 0$, there exist a constant $\rho > 0$ sufficiently large such that for $|x| > \rho$,

Since $\{u_{n}\}$ is bounded and $u_{n}\rightharpoonup 0$ in $E_{\varepsilon}$, by (3.5) we have

On the other hand,

By (2.2), (3.5) and the boundedness of $\{u_{n}\}$, we obtain

From the boundedness of $K(x)$ and $\{u_{n}\}$, there is a constant $C > 0$ such that

By (3.9) and $u_{n}\rightharpoonup 0$, we have

Similarly, we have

By (3.7), (3.8), (3.10) and (3.11), we deduce that

Combining (3.6), (3.12) with (3.4), we obtain

From $u_{n}\not\rightarrow 0$ in $E_{\varepsilon}$, there exists a sequence $\{z_{n}\} \subset \mathbb{R}^{N}$ and constant $R, \beta > 0$ such that

Indeed, if (3.14) does not true, Lemma 2.1 implies that $u_{n}\rightarrow 0$ in $L^{r}(\mathbb{R}^{N})$ for every $r\in (p, p_{s}^{*})$. It follows from (2.2) that

This and (3.2) implies $\|u_{n}\|_{\varepsilon}\rightarrow 0$ as $n\rightarrow \infty$, which contradicts to $u_{n}\not\rightarrow 0$ in $E_{\varepsilon}$.

Now we set $v_{n}(x) = u_{n}(x + z_{n})$. We known that $\{v_{n}\}$ is bounded. Then there exists $v\in W^{s, p}(\mathbb{R}^{N})$ such that $v_{n} \rightharpoonup v$ weakly in $W^{s, p}(\mathbb{R}^{N})$. By (3.14), we see that $v\neq 0$. Hence, there is a set $\Omega \subset \mathbb{R}^{N}$ with $|\Omega| > 0$ such that $v(x) > 0$ in $\Omega$. Then from (3.13) and $t_{n} \geq 1+ \sigma$, we have

Taking limit in the above inequality and by Fatou's lemma, we get

for any $\nu > 0$. It's a contradiction. Therefore, $\limsup_{n\rightarrow \infty} t_{n} \leq 1$.

We next consider the following two cases:

Case 1. $\limsup_{n\rightarrow \infty} t_{n} = 1$. We assume that there exists a subsequence, still denoted by $\{t_{n}\}$ such that $\lim_{n\rightarrow \infty} t_{n} = 1$. Recalling $t_{n}u_{n}\in \mathcal{M}_{V_{\infty}K_{\infty}}$, then

We observe that

From the boundedness of $\{u_{n}\}$, $\lim_{n\rightarrow \infty} t_{n} = 1$, $u_{n} \rightharpoonup 0$ in $E_{\varepsilon}$ and (3.5), one has

and

By (3.12), we have

It follows from (3.15)–(3.19) that

Letting $n\rightarrow \infty$ and $\nu \rightarrow 0$, we get $c \geq c_{\infty}$.

Case 2. $\limsup_{n\rightarrow \infty} t_{n} = t_{0} < 1$. In this case, without loss of generality, we assume that $t_{n} < 1$ for all $n$. Recalling that $t_{n}u_{n}\in \mathcal{M}_{V_{\infty}K_{\infty}}$, then by $I'_{\varepsilon}(u_{n}) \rightarrow 0$, (3.6) and the boundedness of $\{u_{n}\}$, we have

Letting $n\rightarrow \infty$ and $\nu \rightarrow 0$, we get $c_{\infty} \leq c$. The proof is completed.

Lemma 3.2. The functional $I_{\varepsilon}$ satisfies $(PS)_{c}$ condition with $c < c_{\infty}$.

Proof. Let $\{u_{n}\}\subset E_{\varepsilon}$ be a sequence such that $I_{\varepsilon}(u_{n}) \rightarrow c$ and $I'_{\varepsilon}(u_{n}) \rightarrow 0$ as $n \rightarrow \infty$. By (3.1) we have that $\{u_{n}\}$ is bounded in $E_{\varepsilon}$. Then, up to a subsequence, there exists $u\in E_{\varepsilon}$ such that

By (3.20), $p < q < \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N})$, and Hardy-Littlewood-Sobolev inequality, we obtain that

Then, for any $\phi \in C^{\infty}_{0}(\mathbb{R}^{N})$, we have

Then, we have $I'_{\varepsilon}(u) = 0.$ Set $w_{n} = u_{n} -u$. By Brezis-Lieb lemma, we have

From a Brezis-Lieb lemma for the nonlocal term of the functional (^[17]), we obtain

It follows from (3.21) and (3.22) that

and $I'_{\varepsilon}(w_{n}) \rightarrow 0$ as $n \rightarrow \infty$. Since $I'_{\varepsilon}(u) = 0$, we get

Hence, $I_{\varepsilon}(w_{n}) \rightarrow c-I_{\varepsilon}(u) < c_{\infty}$. By Lemma 3.1, $w_{n} \rightarrow 0$ in $E_{\varepsilon}$. Then $u_{n}\rightarrow u$ in $E_{\varepsilon}$. The proof is completed.

Lemma 3.3. Let $\{u_{n}\}$ be a $(PS)_{c}$ sequence restricted in $\mathcal{N}_{\varepsilon}$ and assume that $c < c_{\infty}$. Then $\{u_{n}\}$ has a convergent subsequence in $E_{\varepsilon}$.

Proof. Let $\{u_{n}\}$ be a $(PS)_{c}$ sequence for $I_{\varepsilon}$ on $\mathcal{N}_{\varepsilon}$ at level $c$, namely

It's easy to check that $\{u_{n}\}$ is bounded in $E_{\varepsilon}$. We assume that

where $g(u) = \langle I'_{\varepsilon}(u), u\rangle$, and

Since $\{u_{n}\}$ is bounded, we have

Since $u_{n}\in \mathcal{N}_{\varepsilon}$, by (2.3) and (3.23) we have

where $r^{*}$ is defined in (2.3). Then,

Thus $\lambda_{n} \rightarrow 0$ and $I'_{\varepsilon}(u_{n}) \rightarrow 0$ as $n\rightarrow \infty$. Therefore, $\{u_{n}\}$ is a $(PS)_{c}$ sequence for $I_{\varepsilon}$ in $E_{\varepsilon}$. By Lemma 3.2, $\{u_{n}\}$ has a convergent subsequence.

4.   Proof of Theorem 1.1

We only give the details proof under the assumptions ($\text{H}_{1}$) and ($\text{H}_{2}$). The arguments of ($\text{H}_{3}$) is similar. Under the assumption ($\text{H}_{2}$), we may suppose that $x_{*} = 0\in\mathcal{V}$ or $x_{*} = 0\in\mathcal{V}\cap \mathcal{K}$ if $\mathcal{V}\cap \mathcal{K} \neq \emptyset$. Then

Lemma 4.1. $\limsup_{\varepsilon\rightarrow 0}c_{\varepsilon} \leq c_{\theta\alpha}$. In particular, if $\mathcal{V}\cap \mathcal{K} \neq \emptyset$, then $\limsup_{\varepsilon\rightarrow 0}c_{\varepsilon} = c_{\theta\kappa}$.

Proof. Let $w\in \mathcal{M}_{\theta\alpha}$ be such that

Then there exists a unique $t_{\varepsilon} > 0$ such that $t_{\varepsilon}w \in \mathcal{N}_{\varepsilon}$. Thus

Observe that

Since $t_{\varepsilon}w \in \mathcal{N}_{\varepsilon}$, by the boundedness of $K(x)$, we get that there exist $T_{2} > T_{1} > 0$ such that $T_{1}\leq t_{\varepsilon} < T_{2}$. We may assume that $t_{\varepsilon} \rightarrow t_{0}$ as $\varepsilon \rightarrow 0$. Then by the boundedness of $V$, $K$, and the Lebesgue's theorem, we have

and

Thus, by (4.2) we obtain

It follows from (4.1) that

The proof is completed.

Proposition 1. Assume that ($\mathit{\text{H}}_{1}$) and ($\mathit{\text{H}}_{2}$) hold. Then for any $\varepsilon > 0$ small enough, problem (2.1) has a positive ground state solution.

Proof. Let $\{u_{n}\}$ denotes the (PS) sequence for $I_{\varepsilon}$ given in (2.4). Recall that $\theta < V_{\infty}$ and $\alpha \geq K_{\infty}$. It follows from Lemma 2.2 that $c_{\theta\alpha} < c_{\infty}$. Then, by Lemmas 3.2 and 4.1, we obtain $I_{\varepsilon}$ satisfies the $(PS)_{c_{\varepsilon}}$ condition for $\varepsilon > 0$ small enough. Hence, by mountain pass lemma we have problem (2.1) has a nontrivial ground state solution $u_{\varepsilon}$. We note that all the calculations above can be repeated by word by word, replacing $I^{+}_{\varepsilon}$ with the functional

Then we get a ground state solution $u_{\varepsilon}$ of the equation

Taking $u^{-}_{\varepsilon}$ as a test function in above equation, we have

For any $p\geq 1$, we have

Combining (4.3) with (4.4) yields

Thus, we have $u^{-}_{\varepsilon}(x)\equiv 0$ and $u_{\varepsilon}\geq 0$. It follows from the maximum principle (^[22]) that $u_{\varepsilon} > 0$ in $\mathbb{R}^{N}$.

Lemma 4.2. Let $u_{\varepsilon_{n}}$ be a solution of (2.1) given in Proposition 1. Then, there exists a sequence $\{z_{\varepsilon_{n}}\} \subset \mathbb{R}^{N}$ with $\varepsilon_{n}z_{\varepsilon_{n}} \rightarrow z_{0}\in \Omega_{V}$ such that $v_{\varepsilon_{n}} = u_{\varepsilon_{n}}(x + z_{\varepsilon_{n}})$ converges strongly in $W^{s, p}(\mathbb{R}^{N})$ to a ground state solution of

In particular, if $\mathcal{V}\cap \mathcal{K} \neq \emptyset$, then $z_{0}\in \mathcal{V}\cap \mathcal{K}$, and up to a subsequence, $v_{\varepsilon_{n}}$ converges in $W^{s, p}(\mathbb{R}^{N})$ to a ground state solution of

Proof. Let $\varepsilon_{n} \rightarrow 0$ as $n \rightarrow \infty$, $u_{n} : = u_{\varepsilon_{n}} \in \mathcal{N}_{\varepsilon_{n}}$ be a solution of $(2.1)$. Then $I_{\varepsilon_{n}}(u_{n}) = c_{\varepsilon_{n} }$ and $I'_{\varepsilon_{n}}(u_{n}) = 0$. It is easy to check that $\{u_{n}\}$ is bounded. Then, there exist $R^{*}, \beta > 0$ and a sequence $\{z_{n}\} \subset \mathbb{R}^{N}$ such that

Now we set $v_{n} = u_{n}(x + z_{n})$. Then $v_{n}$ is a solution of the following equation

with the energy

where $V_{n}(x) = V(\varepsilon_{n}x +\varepsilon_{n}z_{n})$ and $K_{n}(x) = K(\varepsilon_{n}x +\varepsilon_{n}z_{n})$. We see that $\{v_{n}\}$ is bounded, then there exists $v \in W^{s, p}(\mathbb{R}^{N})$ satisfying, after passing to a subsequence if necessary

It follows from (4.5) that $v\neq 0$.

We next show that $\{\varepsilon_{n}z_{n}\}$ is bounded. Assume by contradiction that $|\varepsilon_{n}z_{n}|\rightarrow \infty$ as $n \rightarrow \infty$. By the boundedness of $V$ and $K$, we may assume that

By the definition of $V_{\infty}$ and $K_{\infty}$, we have that

Since $V$ and $K$ are uniformly continuous, by (4.8) one has

and

uniformly on bounded sets of $\mathbb{R}^{N}$. Then we have

as $n \rightarrow \infty$ uniformly on bounded sets of $\mathbb{R}^{N}$. From (4.7) and (4.10), for each $\varphi \in C^{\infty}_{0}(\mathbb{R}^{N})$, we have

and

Moreover, by (4.7), (4.10) and Hardy-Littlewood-Sobolev inequality, we infer that

and

Then, for each $\varphi \in C^{\infty}_{0}(\mathbb{R}^{N})$

Since $v_{n}$ satisfies (4.6), by (4.11)–(4.13), for any $\varphi \in C^{\infty}_{0}(\mathbb{R}^{N})$, we have

which implies $v$ is a solution of

By (4.9) and Lemma 2.2, we have

Using Fatou's lemma and Lemma 4.1, we get

which contradicts to (4.14). Hence, $\{\varepsilon_{n}z_{n}\}$ is bounded.

Passing to a subsequence, still denoted by $\{\varepsilon_{n}z_{n}\}$, we may assume that $\varepsilon_{n}z_{n} \rightarrow z_{0}$ as $n \rightarrow \infty$. Then $V_{n}(x)\rightarrow V(z_{0})$ and $K_{n}(x)\rightarrow K(z_{0})$ as $n \rightarrow \infty$. Hence, $v$ is a solution of

Next, we claim that $z_{0}\in \Omega_{V}$. Suppose by contradiction that $z_{0}\not\in \Omega_{V}$, then by ($\text{H}_{2}$) and Lemma 2.2, we have $c_{V(z_{0})K(z_{0})} > c_{\theta\alpha}$. It follows from Lemma 4.1 and the proof of (4.15) that

which is absurd. Hence, $z_{0}\in \Omega_{V}$, and then $\lim_{n\rightarrow \infty}dist(\varepsilon_{n}z_{n}, \Omega_{V}) = 0$. Moreover, $v$ is a ground state solution of (4.16). In particular, if $\mathcal{V}\cap \mathcal{K} \neq \emptyset$, we see that $\lim_{n\rightarrow \infty}dist(\varepsilon_{n}z_{n}, \mathcal{V}\cap \mathcal{K}) = 0$ and $v$ is a ground state solution of

Finally, we shall prove that $v_{n} \rightarrow v$ in $W^{s, p}(\mathbb{R}^{N})$. Since $v$ is a ground state solution of (4.16), by Fatou's lemma, we have

Then

Thus by $V_{n}(x)\rightarrow V(z_{0})$ and Brezis-Lieb lemma, we get $v_{n} \rightarrow v$ in $W^{s, p}(\mathbb{R}^{N})$.

The proof is completed.

Lemma 4.3. $v_{n}\in L^{\infty}(\mathbb{R}^{N})$ and there exists $C > 0$ such that $|v_{n}|_{\infty} \leq C$ for all $n$. Furthermore, $\lim_{|x|\rightarrow \infty} v_{n}(x) = 0$ uniformly in $n$, where $v_{n}$ is given in Lemma 4.2.

Proof. Given $T > 0$ and $\beta > 0$. For each $n$, we denote $v_{T, n} = v_{n}-(v_{n}-T)^{+}$ and $g(v_{n}) = v_{n}v^{p\beta}_{T, n}\in W^{s, p}(\mathbb{R}^{N})$. Taking $g(v_{n})$ as the test function in (4.6), we have

By the boundedness of $K$ and $\{v_{n}\}$, we have for each $n$,

Therefore,

Set

We have for any $a, b \in \mathbb{R}$,

It follows from (4.18) and (4.19) that

By the definition of $g$, we can get $G(v_{n}) \geq \frac{1}{\beta +1}v_{n}v^{\beta}_{T, n}$. From (4.20) and the Sobolev inequality we obtain

Choosing $T_{0} > 1$, by the definition of $v_{T, n}$, we obtain

Since $v_{n} \rightarrow v$ in $W^{s, p}(\mathbb{R}^{N})$, for $T_{0}$ large enough, we conclude that

By (4.22), (4.23) and the boundedness of $\{v_{n}\}$, we get

Letting $T \rightarrow \infty$, by Fatou's lemma, we conclude that $v_{n} \in L^{q+p\beta}(\mathbb{R}^{N})$, and (4.21) implies

Now, from an iterative procedure, in a finite number of steps, we can show that $v_{n}\in L^{\infty}(\mathbb{R}^{N})$ and there exists $C > 0$ such that $\|v_{n}\|_{\infty} \leq C$. Moreover, by $v_{n} \rightarrow v$ in $W^{s, p}(\mathbb{R}^{N})$, we infer that $\lim_{|x|\rightarrow \infty} v_{n}(x) = 0$ uniformly in $n$.

The proof is completed.

Proof of Theorem 1.1. From proposition 1, problem (2.1) has a positive ground state solution $u_{\varepsilon}$. Then the function $w_{\varepsilon}(x) = u_{\varepsilon}(\frac{x}{\varepsilon})$ is a positive ground state solution of (1.1). Now we show the concentration of the maximum points. Let $u_{\varepsilon_{n}}$ be a solution of (2.1). By Lemma 4.2, we have that $v_{\varepsilon_{n}} = u_{\varepsilon_{n}}(x + z_{\varepsilon_{n}})$ is a solution of the problem (4.6). Furthermore, $v_{\varepsilon_{n}} \rightarrow v$ in $W^{s, p}(\mathbb{R}^{N})$ and $\varepsilon_{n}z_{\varepsilon_{n}} \rightarrow z_{0}\in \Omega_{V}$. Furthermore, from Lemma 4.3 we have $v_{n}\in L^{\infty}(\mathbb{R}^{N})$ for all $n$. It follows from (4.5) that

This implies there exist $\iota > 0$ such that

Set $p_{n}$ be a global maximum of $v_{n}$, by Lemma 4.3 and (4.24), we have that $p_{n} \in B_{R}(0)$ for some $R > 0$. Hence, the global maximum of $u_{\varepsilon_{n}}$ given by $y_{\varepsilon_{n}} = p_{n} + z_{\varepsilon_{n}}$. Since $\{p_{n}\}$ is bounded and $\varepsilon_{n}z_{\varepsilon_{n}} \rightarrow z_{0}$, we have $\varepsilon_{n}y_{\varepsilon_{n}} \rightarrow z_{0}$, thus the continuity of $V$ and $K$ gives $\lim_{n\rightarrow \infty} V(\varepsilon_{n}z_{\varepsilon_{n}}) = V(z_{0})$ and $\lim_{n\rightarrow \infty} K(\varepsilon_{n}z_{\varepsilon_{n}}) = K(z_{0})$.

The proof is completed.

5.   Proof of Theorem 1.2

Let $\delta > 0$ be fixed. Define a smooth cut-off function $\eta: \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ satisfying $\eta(t) = 1$ if $0 \leq t \leq \frac{\delta}{2}$, $\eta(t) = 0$ if $t \geq \delta$. For any $\xi\in \Lambda$, define

where $\omega(x)$ is the positive ground state solution of (4.17). By definition, $W_{\varepsilon, \xi}$ has compact support for any $\xi\in \Lambda$, and hence it belongs to $E_{\varepsilon}$. It's easy to check that there exists a unique $t_{\varepsilon} > 0$ such that $t_{\varepsilon}W_{\varepsilon, \xi} \in \mathcal{N}_{\varepsilon}$. Now, we define the map $\phi_{\varepsilon}: N\rightarrow \mathcal{N}_{\varepsilon}$ by $\phi_{\varepsilon}(\xi) = t_{\varepsilon}W_{\varepsilon, \xi}$.

Lemma 5.1. Uniformly in $\xi\in \Lambda$, we have

Proof. Let $\xi\in \Lambda$. By the definition of $\phi_{\varepsilon}(\xi)$ and a simple change of variable, one has

By Lemma 2.2 of ^[2] and Lebesgue's theorem, we get

and

On the other hand, since $\omega$ is a ground state solution of (4.17), we have

Then, by (5.1)–(5.5), we deduce that $\lim_{\varepsilon \rightarrow 0}t_{\varepsilon} = 1$. At this point, by the same change of variable as before and (5.2), (5.3), we have

Moreover, the limit is uniformly in $\xi$.

The proof is completed.

Let $R > 0$ be such that $\Lambda_{\delta} \subset B_{R}(0)$. Define $\chi: \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ by $\chi(x) = x$ for $x\in B_{R}(0)$ and $\chi(x) = \frac{Rx}{|x|}$ for $x\in \mathbb{R}^{N}\backslash B_{R}(0)$. Now we define $\rho_{\varepsilon}: \mathcal{N}_{\varepsilon} \rightarrow \mathbb{R}^{N}$ by

By the definition of $\chi$ and the Lebesgue's theorm, we have that

uniformly for $\xi \in \Lambda_{\delta}$.

Let $f(\varepsilon)$ be any positive function tending to $0$ as $\varepsilon \rightarrow 0$. Set

By Lemma 5.1, we see that $\overline{\mathcal{N}}_{\varepsilon}\neq \emptyset$ for $\varepsilon > 0$ small enough.

Lemma 5.2.

Proof. Let $\varepsilon_{n} \rightarrow 0$ as $n \rightarrow \infty$, by the definition, there exists a sequence $\{u_{n}\} \subset \overline{\mathcal{N}}_{\varepsilon_{n}}$ such that

So it suffices to find a sequence $\{\xi_{n}\} \subset \Lambda_{\delta}$ satisfying

Since $\{u_{n}\} \subset \overline{\mathcal{N}}_{\varepsilon_{n}} \subset \mathcal{N}_{\varepsilon_{n}}$, by Lemma 2.2 we have that

this implies $I_{\varepsilon_{n}}(u_{n}) \rightarrow c_{\theta\kappa}$ as $n \rightarrow \infty$. By the proof of Lemma 4.2, we obtain that there exists a sequence $\{ \overline{\xi_{n}}\} \subset \mathbb{R}^{N}$ such that $\varepsilon_{n}\overline{\xi_{n}} \rightarrow \overline{z_{0}} \in \Lambda$ and $v_{n}(x) = u_{n}(x+\overline{\xi_{n}})$ converges strongly in $W^{s, p}(\mathbb{R}^{N})$ to $v$, a positive ground state of (4.17). Set $\xi_{n} = \varepsilon_{n}\overline{\xi_{n}}$. Then by Lebesgue's theorem,

Thus (5.7) holds.

The proof is completed.

Proof of Theorem 1.2. For a fixed $\delta > 0$, by Lemma 5.1 there exists $\varepsilon_{\delta} > 0$ such that $I_{\varepsilon}(\phi_{\varepsilon}(\xi))\leq c_{\theta\kappa} + f(\varepsilon)$ for any $\varepsilon \in (0, \varepsilon_{\delta})$ and $\xi \in \Lambda$. Using Lemma 5.2, we have $dist(\rho_{\varepsilon}(u), \Lambda_{\delta}) < \frac{\delta}{2}$ for such $\varepsilon$ and $u\in \overline{\mathcal{N}}_{\varepsilon}$. It follows that the map $\rho_{\varepsilon}\circ \phi_{\varepsilon}: \Lambda\rightarrow \Lambda_{\delta}$ is well defined. Then, by (5.6) the map $\rho_{\varepsilon}\circ \phi_{\varepsilon}$ is homotopic to the inclusion map: $Id: \Lambda\rightarrow \Lambda_{\delta}$. Applying homotopic and by the same arguments of ^[6], we obtain that $cat_{\overline{\mathcal{N}_{\varepsilon}}}(\overline{\mathcal{N}_{\varepsilon}}) \geq cat_{\Lambda_{\delta}}(\Lambda)$. By the definition of $\overline{\mathcal{N}_{\varepsilon}}$ and choosing $\varepsilon_{\delta}$ small, from Lemma 3.3, we have that $I_{\varepsilon}$ satisfies the $(PS)$ condition in $\overline{\mathcal{N}_{\varepsilon}}$. Therefore, standard Ljusternik-Schnirelmann theory implies that $I_{\varepsilon}$ has at least $cat_{\overline{\mathcal{N}_{\varepsilon}}}(\overline{\mathcal{N}_{\varepsilon}})$ critical points on $\mathcal{N}_{\varepsilon}$. It is easy to check that $I_{\varepsilon}$ has at least $cat_{\Lambda_{\delta}}(\Lambda)$ critical points in $E_{\varepsilon}$. The concentration behavior of these solutions as $\varepsilon \rightarrow 0$ are similar as in the proof of Theorem 1.1.

Acknowledgments

The authors would like to express sincere thanks to the anonymous referees for their carefully reading this paper and valuable useful comments. This work was supported by NSFC (No.11601234, 11571176), Natural Science Foundation of Jiangsu Province of China for Young Scholar (No.BK20160571).

Conflict of interest

The author declares no conflicts of interest in this paper.