Existence and concentration of ground state solutions for a class of fractional Schrödinger equations

Abstract

In this paper, by using variational methods, we study the existence and concentration of ground state solutions for the following fractional Schrödinger equation $\begin{matrix} {(- Δ)}^{α} u + V (x) u = A (ϵ x) f (u), x \in R^{N}, \end{matrix}$ where $α \in (0, 1)$ , ϵ is a positive parameter, $N > 2 α$ , ${(- Δ)}^{α}$ stands for the fractional Laplacian, f is a continuous function with subcritical growth, $V \in C (R^{N}, R)$ is a $Z^{N}$ -periodic function and $A \in C (R^{N}, R)$ satisfies some appropriate assumptions.

Keywords

Concentration of solutions ground state fractional Schrödinger equations periodic potential Nehari manifold

1. Introduction

In this paper, we are interested in the following fractional Schrödinger equation $\begin{matrix} (P_{ϵ}) & \{\begin{matrix} {(- Δ)}^{α} u + V (x) u = A (ϵ x) f (u), x \in R^{N}, \\ u \in H^{α} (R^{N}), \end{matrix} \end{matrix}$ where $α \in (0, 1)$ , $ϵ > 0$ is a parameter, $N > 2 α$ , ${(- Δ)}^{α}$ stands for the fractional Laplacian operator, f is a continuous function with subcritical growth, the potential $V \in C (R^{N}, R)$ is a periodic function verifying $\begin{matrix} (V) & 0 \notin σ ({(- Δ)}^{α} + V) and σ ({(- Δ)}^{α} + V) \cap (- \infty, 0) \neq \emptyset, \end{matrix}$ where $σ ({(- Δ)}^{α} + V)$ denotes the spectrum of ${(- Δ)}^{α} + V$ in $L^{2} (R^{N})$ . $A \in C (R^{N}, R)$ satisfies the following condition $\begin{matrix} (A) & 0 < inf_{x \in R^{N}} A (x) ⩽ lim_{| x | \to + \infty} A (x) < sup_{x \in R^{N}} A (x) . \end{matrix}$

When $α = 1$ , the problem ( P ϵ ) is a classical semilinear elliptic equation. In recent years, the existence and concentration of nontrival solutions for semilinear elliptic equations involving in different nonlinearities and potential functions have been widely studied, such as [1,2,5,6,17,20,25,26,32] and their references. Due to our scope, we would like to mention some works related of the Schrödinger equations with periodic data. Coti Zelati and Rabinowitz [11] considered this kind of problems in the definite case $0 < inf (- Δ + V)$ . For the indefinite case, we main consider the situation that 0 lies in the spectrum gap of $- Δ + V$ , thus the space which was spanned by the eigenfunctions corresponding of negative eigenvalue of the operator $- Δ + V$ in is infinite dimensional, it is easy to see that zero is no longer a local minimum point of the corresponding energy functional, thus the usual mountain pass geometry does not work and the classical linking theorem or fountain theorem can not be used to this problem. To overcome this difficulty, there are some different methods have been proposed in the last twenty years. Li and Szulkin [18] first developed a new generalized linking theorem, after that, Szulkin and Weth [31] studied the ground state solutions based on the generalized Nehari manifold or called Nehari–Pankov manifold [23] $\begin{matrix} M : = {u \in E ∖ E^{-} : J^{'} (u) u = 0 and J^{'} (u) v = 0 for all v \in E^{-}}, \end{matrix}$ they constructed a new minimax characterization such that an indefinite problem can be reduced to a definite problem. Meanwhile, Liu [19] proved the existence of ground state solutions by generalized linking theorem under a more weakly monotone condition, and he used super-quadratic condition to replace the standard Ambrosetti–Rabinowitz condition. Recently, Alves and Germano [1] studied a class of semilinear equations $\begin{matrix} (1.1) & - Δ u + V (x) u = A (ϵ x) f (u), x \in R^{N}, \end{matrix}$ where the function $V (x)$ and $A (x)$ satisfy the conditions ( V ), ( A ), $ϵ > 0$ is a parameter, f is continuous and satisfies some assumptions, the authors considered the existence and concentration of ground states in subcritical case and critical case, respectively.

In the past several decades, the fractional Schrödinger equations has been widely studied. We refer to Molica Bisci, Rădulescu and Servadei [22] who carefully studied continuous and compact embeddings, trace theory, extension, and regularity in fractional Sobolev space. For the strongly indefinite problem, Fang and Ji [12], Pu et al. [24] used different methods to prove the existence of ground state solutions with different assumptions on the nonlinearity. For more research related fractional PDEs, we refer to [3,4,7,9,10,13–16,21,22,26–28,33,34] and the references therein.

Inspired by [1 ,31], we would like to study the existence and concentration of ground state solutions for the problem ( P ϵ ). Since the fractional Laplacian operator is a nonlocal operator, our problem becomes more difficult and the $L^{\infty}$ -estimate of ground state solutions is more complicated than the local equations.

From now on, we assume the following hypotheses on the nonlinearity f:

${lim}_{t \to 0} \frac{f (t)}{t} = 0$ ;

${lim sup}_{| t | \to + \infty} \frac{f (t)}{| t |^{q}} < + \infty$ for $q \in (1, 2_{α}^{*} - 1)$ , $2_{α}^{*} = \frac{2 N}{N - 2 α}$ is called the fractional critical exponent;

$t \mapsto f (t) / | t |$ is strictly increasing on $(0, + \infty)$ ;

There exists $μ > 2$ such that $\begin{matrix} 0 < μ F (t) ⩽ f (t) t, \forall t \in R ∖ {0}, \end{matrix}$ where $\begin{matrix} F (t) = \int_{0}^{t} f (x) d x . \end{matrix}$

We are now in position to give the main result of this paper.

Theorem 1.1.
Assume that ( V ), ( A ) and $(f_{1})$ – $(f_{4})$ hold. Then, there exists $ϵ_{0} > 0$ such that the problem ( P ϵ ) has a ground state solution for each $ϵ \in (0, ϵ_{0})$ . Furthermore, if $x_{ϵ} \in R^{N}$ denotes a global maximum point of $| u_{ϵ} |$ , then $\begin{matrix} lim_{ϵ \to 0} A (ϵ x_{ϵ}) = sup_{x \in R^{N}} A (x) . \end{matrix}$

This paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we prove some results about the autonomous problems. In Section 4, the existence of ground state solutions for the problem ( P ϵ ) is proved. The last Section is devoted to prove the concentration result in Theorem 1.1.
2. Preliminaries

In this section, we introduce the setting framework of the fractional Sobolev space. The fractional Laplace operator ${(- Δ)}^{α} u$ can be given by the following singular integral $\begin{matrix} {(- Δ)}^{s} u = c_{N, s} P . V . \int_{R^{N}} \frac{u (x) - u (y)}{| x - y |^{N + 2 s}} d y, \end{matrix}$ where $P . V .$ is the Cauchy principal value and $c_{N, s} = π^{(- 2 s + \frac{N}{2})} \frac{Γ (\frac{N}{2} + s)}{Γ (- s)}$ is a normalization constant, or it also can be defined on the Schwartz class of function $u \in S (R^{N})$ through the Fourier transform $F$ $\begin{matrix} F ({(- Δ)}^{s} u) (ξ) = | ξ |^{2 s} F u (ξ) . \end{matrix}$

Next, we define the fractional Sobolev space $E : = H^{α} (R^{N})$ as follow $\begin{matrix} H^{α} (R^{N}) = {u \in L^{2} (R^{N}) : \frac{| u (x) - u (y) |}{| x - y |^{\frac{N}{2} + α}} \in L^{2} (R^{N} \times R^{N})}, \end{matrix}$ endowed with the natural norm $\begin{matrix} (2.1) & ‖ u ‖_{0} = {(\iint_{R^{N} \times R^{N}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 α}} d x d y + \int_{R^{N}} u^{2} d x)}^{\frac{1}{2}} . \end{matrix}$

The variational functional associated with the problem ( P ϵ ) is $\begin{matrix} (2.2) & I_{ϵ} (u) = \frac{1}{2} {(\iint_{R^{N} \times R^{N}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 α}} d x d y + \int_{R^{N}} V (x) u^{2} d x)}^{\frac{1}{2}} - \int_{R^{N}} A (ϵ x) F (u) d x . \end{matrix}$ It’s routine to show that $I_{ϵ} (u) \in C^{1} (H^{α} (R^{N}), R)$ .

Let’s define an operator $\begin{matrix} L : = {(- Δ)}^{α} + V (x) . \end{matrix}$ Since $V (x)$ is a continuous periodic function, thus $V (x)$ is bounded in $R^{N}$ . Then it is easy to prove that L is a self-adjoint operator. Let $| L |^{\frac{1}{2}}$ be the square root of the absolute value of L on $H^{α} (R^{N})$ , then we choose the inner product $\begin{matrix} a (u, v) = \int_{R^{N}} | L |^{\frac{1}{2}} u | L |^{\frac{1}{2}} v d x, u, v \in H^{α} (R^{N}), \end{matrix}$ with the norm $\begin{matrix} ‖ u ‖ : = a {(u, u)}^{\frac{1}{2}} . \end{matrix}$

We define a bilinear map $B (u, v)$ with the following form $\begin{matrix} (2.3) & B (u, v) = \iint_{R^{N} \times R^{N}} \frac{(u (x) - u (y)) (v (x) - v (y))}{| x - y |^{N + 2 α}} d x d y + \int_{R^{N}} V (x) u (x) v (x) d x . \end{matrix}$ There exists an orthogonal decomposition $E : = E^{+} \oplus E^{-}$ such that $u = u^{+} + u^{-}$ and $B (u, u) = B (u^{+}, u^{+}) + B (u^{-}, u^{-})$ for each $u \in H^{α} (R^{N})$ . Let $B (u^{+}, u^{+}) = a (u^{+}, u^{+})$ and $B (u^{-}, u^{-}) = - a (u^{-}, u^{-})$ for all $u \in H^{α} (R^{N})$ . Thus, the functional (2.2) is equivalent to the following form $\begin{matrix} I_{ϵ} (u) = \frac{1}{2} ({‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2}) - \int_{R^{N}} A (ϵ x) F (u) d x, \end{matrix}$ which also belongs to $C^{1} (H^{α} (R^{N}), R)$ , and it’s obviously that the critical points of the functional $I_{ϵ} (u)$ correspond to weak solutions of the problem ( P ϵ ).

The following two propositions will be used frequently in the below.

Proposition 2.1 ([26]).

Let $α \in (0, 1)$ and $N > 2 α$ , there exists a constant $C = C (α, N) > 0$ such that $\begin{matrix} ‖ u ‖_{L^{2_{α}^{*}} (R^{N})} ⩽ C ‖ u ‖_{H^{α} (R^{N})}, \forall u \in H^{α} (R^{N}) . \end{matrix}$ Furthermore, the embedding $H^{α} (R^{N}) ↪ L^{p} (R^{N})$ is continuous for each $p \in [2, 2_{α}^{*}]$ , and is locally compact for all $p \in [2, 2_{α}^{*})$ .

Proposition 2.2 ([13,26]).

Assume that the sequence $(u_{n})$ is bounded in $H^{α} (R^{N})$ , and for each $R > 0$ , it satisfies $\begin{matrix} lim_{n \to \infty} sup_{y \in R^{N}} \int_{B_{R} (y)} {| u_{n} (x) |}^{2} d x = 0 . \end{matrix}$ Then $u_{n} \to 0$ in $L^{p} (R^{N})$ for all $p \in (2, 2_{α}^{*})$ .

3. An autonomous problem

In this section, we consider the following autonomous problem $\begin{matrix} (P_{λ}) & \{\begin{matrix} {(- Δ)}^{α} u + V (x) u = λ f (u) in R^{N}, \\ u \in H^{α} (R^{N}), \end{matrix} \end{matrix}$ where $λ > 0$ , $α \in (0, 1)$ , $N > 2 α$ and the conditions ( V ) and $(f_{1})$ – $(f_{4})$ hold. The energy functional $J_{λ}$ associated with the problem ( P λ ) is $\begin{matrix} J_{λ} (u) = \frac{1}{2} \int_{R^{N}} ({| {(- Δ)}^{\frac{α}{2}} u |}^{2} + V (x) u^{2}) d x - λ \int_{R^{N}} F (u) d x, \end{matrix}$ or equivalently $\begin{matrix} J_{λ} (u) = \frac{1}{2} {‖ u^{+} ‖}^{2} - \frac{1}{2} {‖ u^{-} ‖}^{2} - λ \int_{R^{N}} F (u) d x . \end{matrix}$ By the assumptions $(f_{1})$ – $(f_{2})$ , $J_{λ}$ belongs to $C^{1} (H^{α} (R^{N}), R)$ , and the critical points of functional $J_{λ} (u)$ correspond to the weak solutions of the problem ( P λ ).

In order to get the ground state solutions of the problem ( P λ ), we shall apply the method of Nehari manifold developed by Szulkin and Weth [31]. Following the same variational framework as in [31], we have the similar results in fractional Sobolev space.

First of all, we define the ground state energy $d_{λ}$ of functional $J_{λ}$ $\begin{matrix} (3.1) & d_{λ} = inf_{u \in N_{λ}} J_{λ} (u), \end{matrix}$ where $N_{λ}$ is Nehari–Pankov manifold associated with $J_{λ}$ $\begin{matrix} (3.2) & N_{λ} = {u \in H^{α} (R^{N}) ∖ E^{-}; J_{λ}^{'} (u) u = 0 and J_{λ}^{'} (u) v = 0, \forall v \in E^{-}} . \end{matrix}$ Moreover, for each $u \in H^{α} (R^{N}) ∖ E^{-}$ , we introduce two important sets which have been studied in [31], $\begin{matrix} E (u) = E^{-} \oplus R u = E^{-} \oplus R u^{+}, \end{matrix}$ and $\begin{matrix} \hat{E} (u) = E^{-} \oplus [0, + \infty) u = E^{-} \oplus [0, + \infty) u^{+}, \end{matrix}$ where $E (u)$ is a subspace in $H^{α} (R^{N})$ and $\hat{E} (u)$ is a convex subset in $H^{α} (R^{N})$ . Both of them do not depend on λ, which just rely on the operator ${(- Δ)}^{α} + V$ , this is a crucial point in the following proofs.

Lemma 3.1 ([31]).

If $u \in N_{λ}$ , then $\begin{matrix} J_{λ} (u + v) < J_{λ} (u) for every v \in M : = {s u + m : s ⩾ - 1, m \in E^{-}}, v \neq 0 . \end{matrix}$ Hence u is a unique global maximum of $J_{λ} |_{\hat{E} (u)}$ .

Proof.
(2.3) shows that $B : E \times E \to R$ is a symmetric bilinear map. Then we can use the similar method in [31, Proposition 2.3] and condition $(f_{3})$ to get this conclusion. □
Lemma 3.2.

There exists $κ > 0$ such that $c = {inf}_{N_{λ}} J_{λ} ⩾ {inf}_{S_{κ}} J_{λ} > 0$ , where $S_{κ} : = {u \in E^{+} : ‖ u ‖ = κ}$ ;

$‖ u^{+} ‖ ⩾ max {‖ u^{-} ‖, \sqrt{2 c}}$ for all $u \in N_{λ}$ .

Proof.
(a) For any $u \in E^{+}$ , then we have $J_{λ} = \frac{1}{2} ‖ u^{+} ‖^{2} - λ \int_{R^{N}} F (u) d x$ . By $(f_{1})$ , we know that $λ \int_{R^{N}} F (u) d x = o (‖ u ‖^{2})$ as $u \to 0$ . Hence if $‖ u ‖ = κ$ small enough, we have ${inf}_{S_{κ}} J_{λ} > 0$ . Then we prove ${inf}_{N_{λ}} J_{λ} ⩾ {inf}_{S_{κ}} J_{λ}$ . For each $u \in N_{λ}$ , there exists $s > 0$ such that $s u^{+} = κ$ , then $s u^{+} \in \hat{E} (u) \cap S_{κ}$ . By Lemma 3.1, $J_{λ} (u) = {max}_{w \in \hat{E} (u)} J_{λ} (w) ⩾ J_{λ} (s u^{+})$ , thus ${inf}_{N_{λ}} J_{λ} ⩾ {inf}_{S_{κ}} J_{λ}$ .

(b) For every $u \in N_{λ}$ , by the definition of c, we have $\begin{matrix} 0 < c ⩽ \frac{1}{2} {‖ u^{+} ‖}^{2} - \frac{1}{2} {‖ u^{-} ‖}^{2} - λ \int_{R^{N}} F (u) ⩽ \frac{1}{2} {‖ u^{+} ‖}^{2} - \frac{1}{2} {‖ u^{-} ‖}^{2}, \end{matrix}$ where $F (u) ⩾ 0$ by $(f_{4})$ , then we obtain that $‖ u^{+} ‖ ⩾ max {‖ u^{-} ‖, \sqrt{2 c}}$ . □
Lemma 3.3 ([1, Lemma 2.2]).

If $V \subset E^{+} ∖ {0}$ is a compact subset and $0 \notin V^{σ (H^{α} (R^{N}), H^{α} {(R^{N})}^{'})}$ . $P (x) \in C (R^{N}) \cap L^{\infty} (R^{N}) with {inf}_{x \in R^{N}} P (x) = P (0) > 0$ . Otherwise, from $(f_{4})$ , we have the following properties for $F (u)$ ,

$\frac{F (t)}{t^{2}} \to + \infty$ as $| t | \to + \infty$ ;

$F (t) ⩾ 0$ for all $t \in R$ .

Then, for the following functional

φ : H^{α} (R^{N}) \to R \cup {- \infty}

\begin{matrix} φ (u) = \frac{1}{2} {‖ u^{+} ‖}^{2} - \frac{1}{2} {‖ u^{-} ‖}^{2} - \int_{R^{N}} P (x) F (u) d x, \end{matrix}

there exists

R > 0

such that

φ (u) ⩽ 0

E (u) ∖ B_{R} (0)

for every

u \in V

Lemma 3.4.
$J_{λ}$ is coercive on $N_{λ}$ for every $λ > 0$ .
Proof.
Suppose by contradiction, assume that there exists a sequence ${u_{n}} \subset N_{λ}$ such that $J_{λ} (u_{n}) ⩽ C$ for some $C \in (0, + \infty)$ as $‖ u_{n} ‖ \to + \infty$ . Define the sequence $ω_{n} : = \frac{u_{n}}{‖ u_{n} ‖}$ , from Lemma 3.2, we get $‖ u_{n}^{+} ‖ ⩾ ‖ u_{n}^{-} ‖$ , hence $‖ ω_{n}^{+} ‖^{2} ⩾ ‖ ω_{n}^{-} ‖^{2}$ and $‖ ω_{n}^{+} ‖^{2} ⩾ \frac{1}{2}$ for each $n \in N$ . There exists a sequence $(y_{n}) \subset Z^{N}$ , $R > 0$ and $δ > 0$ such that $\begin{matrix} (3.3) & \int_{B_{R} (y_{n})} {| ω_{n}^{+} |}^{2} d x ⩾ δ, \forall n \in N . \end{matrix}$ If not, from Proposition 2.2, we get that $ω_{n}^{+} \to 0$ in $L^{p} (R^{N})$ for $p \in (2, 2_{α}^{})$ . Then $\int_{R^{N}} F (s ω_{n}^{+}) d x \to 0$ for each $s > 0$ by $(f_{1})$ , and $\begin{array}{l} C & ⩾ J_{λ} (s w_{n}^{+}) = \frac{1}{2} s^{2} {‖ ω_{n}^{+} ‖}^{2} - λ \int_{R^{N}} F (s ω_{n}^{+}) d x \\ ⩾ \frac{s^{2}}{4} - λ \int_{R^{N}} F (s ω_{n}^{+}) d x \to \frac{s^{2}}{4}, \end{array}$ which is impossible because s is arbitrary, so (3.3) holds.

Next, we define ${\tilde{u}}_{n} (x) : = u_{n} (x + y_{n})$ and ${\tilde{ω}}_{n} (x) : = ω_{n} (x + y_{n})$ . It’s easy to prove that if each element $u^{+} (x) \in E^{+}$ , then $u^{+} (x + y_{n}) \in E^{+}$ . By (3.3), we obtain that ${\tilde{ω}}_{n}^{+} ⇀ {\tilde{ω}}^{+} \neq 0$ . Because ${\tilde{u}}_{n} (x) = {\tilde{ω}}_{n} (x) ‖ {\tilde{u}}_{n} ‖$ , hence we have ${\tilde{u}}_{n} (x) \to + \infty$ a.e. in $R^{N}$ as $‖ {\tilde{u}}_{n} ‖ = ‖ u_{n} ‖ \to + \infty$ . By the Fatou’s lemma, $\begin{matrix} \int_{R^{N}} \frac{F (u_{n})}{‖ u_{n} ‖^{2}} d x = \int_{R^{N}} \frac{F ({\tilde{u}}_{n})}{| {\tilde{u}}_{n} |^{2}} | {\tilde{ω}}_{n} |^{2} d x ⩾ \int_{[{\tilde{u}}_{n} \neq 0]} \frac{F ({\tilde{u}}_{n})}{| {\tilde{u}}_{n} |^{2}} | {\tilde{ω}}_{n} |^{2} d x \to + \infty, \end{matrix}$ where $[{\tilde{u}}_{n} \neq 0]$ means the Lebesgue measure of the set ${x \in R^{N} : {\tilde{u}}_{n} (x) \neq 0}$ . Thereby, $\begin{array}{l} 0 & ⩽ \frac{J_{λ} (u_{n})}{‖ u_{n} ‖^{2}} = \frac{1}{2} {‖ ω_{n}^{+} ‖}^{2} - \frac{1}{2} {‖ ω_{n}^{-} ‖}^{2} - λ \int_{R^{N}} \frac{F (u_{n})}{‖ u_{n} ‖^{2}} d x \\ ⩽ \frac{1}{2} - λ \int_{R^{N}} \frac{F ({\tilde{u}}_{n})}{| {\tilde{u}}_{n} |^{2}} | {\tilde{ω}}_{n} |^{2} d x \to - \infty, \end{array}$ which is absurd. □
Lemma 3.5.
For every* $u \in H^{α} (R^{N}) ∖ E^{-}$ , the set $N_{λ} \cap \hat{E} (u)$ has a unique element $m_{λ} (u)$ , and it is the global maximum of $J_{λ} |_{\hat{E} (u)}$ .
Proof.
From Lemma 3.1, we know that $N_{λ} \cap \hat{E} (u) \neq \emptyset$ . Let $u \in E^{+}$ and $‖ u ‖ = 1$ . By Lemma 3.2(a), $J_{λ} (t u) > 0$ for $t \in (0, + \infty)$ small enough, and $J_{λ} (t u) ⩽ 0$ for $t u \in \hat{E} (u) ∖ B_{R} (0)$ by Lemma 3.3, which means that ${sup}_{w \in \hat{E} (u)} J_{λ} (w) > 0$ . Since $\hat{E} (u)$ is a convex set, and the functional is weakly supper semi-continuous on $\hat{E} (u)$ , by [30, Theorem 1.2], we can deduce that ${sup}_{w \in \hat{E} (u)} J_{λ} (w)$ is attained, and there exists $u_{0} \in \hat{E} (u) ∖ {0}$ such that $J_{λ} (u_{0}) = {sup}_{w \in \hat{E} (u)} J_{λ} (w)$ . It shows that $u_{0}$ is a critical point of $J_{λ} |_{\hat{E} (u)}$ , so it satisfies $< J_{λ}^{'} (u_{0}), u_{0} > = < J_{λ}^{'} (u_{0}), v > = 0$ for all $v \in \hat{E} (u)$ , which means that $u_{0} \in N_{λ}$ . Therefore, $u_{0} \in N_{λ} \cap \hat{E} (u)$ . □
Remark 3.6.
From the last lemma, we know that for each $u \in H^{α} (R^{N}) ∖ E^{-}$ , there exists a unique pair $(t, v)$ , $t \in (0, + \infty)$ and $v \in E^{-}$ such that $t u + v \in N_{λ} \cap \hat{E} (u)$ and $\begin{matrix} (3.4) & J_{λ} (t u + v) = max_{w \in \hat{E} (u)} J_{λ} (w) . \end{matrix}$
Lemma 3.7 ([31]).

Assume that ( V ), ( A ) and $(f_{1})$ – $(f_{4})$ hold, then

The map $Ψ_{λ} : u \to m_{λ} (u)$ in $u \in E^{+} ∖ {0}$ is continuous.

$Ψ_{λ} \in C^{1} (E^{+} ∖ {0}, R)$ and $Ψ_{λ}^{'} (u) v = \frac{‖ m_{λ} {(u)}^{+} ‖}{‖ u ‖} J_{λ}^{'} (m_{λ} (u)) v$ where $u, v \in E^{+}$ and $u \neq 0$ for every $λ > 0$ .

Define the map ${\hat{m}}_{λ} (u) : N_{λ} \to S^{+}$ , ${\hat{m}}_{λ} (u) = \frac{u^{+}}{‖ u^{+} ‖}$ and ${\hat{Ψ}}_{λ} : S^{+} \to R$ is $Ψ_{λ} |_{S^{+}}$ . Then ${\hat{Ψ}}_{λ} \in C^{1} (S^{+})$ and $\begin{matrix} {\hat{Ψ}}_{λ}^{'} (u) v = ‖ m_{λ} {(u)}^{+} ‖ J_{λ}^{'} (m_{λ} (u)) v, \end{matrix}$ where $v \in T_{u} S^{+} = {w \in E^{+} : ⟨ v, w ⟩ = 0}$ .

${(u_{n})}_{n}$ is a $(PS)$ -sequence for ${\hat{Ψ}}_{λ}$ if and only if ${(m_{λ} (u_{n}))}_{n}$ is a $(PS)$ -sequence for $J_{λ}$ .

${inf}_{S^{+}} {\hat{Ψ}}_{λ} = {inf}_{N_{λ}} J_{λ} = d_{λ}$ . Furthermore, u is a critical point for ${\hat{Ψ}}_{λ}$ if and only if $m_{λ} (u)$ is a critical point for $J_{λ}$ and ${\hat{Ψ}}_{λ} (u) = J_{λ} (m_{λ} (u))$ .

The problem ( P λ ) has a ground state solution for every $λ > 0$ .

Next we start studying the property of the following function, $\begin{matrix} h (λ) : λ \mapsto d_{λ}, λ \in (0, + \infty), \end{matrix}$ where $d_{λ}$ is the ground state energy for $J_{λ}$ . Define $u_{λ}$ is a ground state solution of ( P λ ), then $u_{λ} \in N_{λ}$ and $d_{λ} = {inf}_{N_{λ}} J_{λ} = J_{λ} (u_{λ})$ , and from Lemma 3.5, it can be defined by $\begin{matrix} (3.5) & 0 < d_{λ} = inf_{u \in E^{+} ∖ {0}} max_{v \in \hat{E} (u)} J_{λ} (v) . \end{matrix}$

Proposition 3.8.
The function $h (λ)$ is decreasing and continuous.
Proof.
First we consider the monotonicity of $h (λ)$ , assume that $u_{λ}$ , $u_{θ}$ are the ground state solution of functional $J_{λ}$ and $J_{θ}$ , respectively. Suppose the function $h (λ)$ is not decreasing, without loss of generality, we assume that $λ > θ$ such that $h (λ) ⩾ h (θ)$ . By (3.4), there exist $t_{λ} > 0$ and $v_{λ} \in E^{-}$ such that $\begin{matrix} J_{λ} (t_{λ} u_{θ} + v_{λ}) = max_{u \in \hat{E} (u_{θ})} J_{λ} (u) . \end{matrix}$ Hence $\begin{array}{l} d_{λ} & ⩽ J_{λ} (t_{λ} u_{θ} + v_{λ}) \\ = J_{θ} (t_{λ} u_{θ} + v_{λ}) + (θ - λ) \int_{R^{N}} F (t_{λ} u_{θ} + v_{λ}) d x \\ ⩽ J_{θ} (u_{θ}) + (θ - λ) \int_{R^{N}} F (t_{λ} u_{θ} + v_{λ}) d x \\ ⩽ d_{θ} + (θ - λ) \int_{R^{N}} F (t_{λ} u_{θ} + v_{λ}) d x . \end{array}$ But from the above assumption, we have $\begin{matrix} (θ - λ) \int_{R^{N}} F (t_{λ} u_{θ} + v_{λ}) d x ⩽ 0, \end{matrix}$ together with $θ - λ < 0$ and condition $(f_{4})$ , we conclude that $\begin{matrix} \int_{R^{N}} F (t_{λ} u_{θ} + v_{λ}) d x = 0 . \end{matrix}$ From $(f_{1})$ , $(f_{4})$ , we know $F (u) > 0$ if $u \neq 0$ and $F (u) \to 0$ as $u \to 0$ . We deduce that $t_{λ} u_{θ} + v_{λ} = 0$ a.e. in $R^{N}$ . Thus $0 = J_{λ} (t_{λ} u_{θ} + v_{λ}) ⩾ d_{λ}$ , which is impossible because $d_{λ} > 0$ . Then we show that the function $h (λ)$ is strictly decreasing.

Next we shall show that the function $h (λ)$ is continuous. To achieve this goal, we want to prove ${lim}_{n \to + \infty} d_{λ_{n}} = d_{λ}$ by two cases, an increasing sequence and a decreasing sequence. Because for any sequences $(λ_{n})$ always can be divided by an increasing sequence and a decreasing sequence, thus if this two cases both hold, then we could know that the function $h (λ)$ is continuous.

The increasing case: Choosing the sequence $(λ_{n})$ with $λ_{1} ⩽ λ_{2} ⩽ \dots ⩽ λ_{n} \to λ$ . From the above result, we have $d_{λ} ⩽ d_{λ_{n}} ⩽ d_{λ_{n - 1}} ⩽ \dots ⩽ d_{λ_{1}}$ , we need to prove that ${lim}_{n \to + \infty} d_{λ_{n}} = d_{λ}$ . Let $u_{λ}$ be the ground state solution of ( P λ ). For each $n \in N$ , there exist $t_{n}$ and $v_{n} \in E^{-}$ such that $\begin{matrix} J_{λ_{n}} (t_{n} u_{λ} + v_{n}) = max_{u \in \hat{E} (u_{λ})} J_{λ_{n}} (u) . \end{matrix}$ Furthermore, $\begin{matrix} J_{λ_{1}} (u) - J_{λ_{n}} (u) = (λ_{n} - λ_{1}) \int_{R^{N}} F (u) d x ⩾ 0, \end{matrix}$ then $J_{λ_{1}} (u) ⩾ J_{λ_{n}} (u)$ for each $n \in N$ and $u \in H^{α} (R^{N})$ . By Lemma (3.3), there exists $R > 0$ such that $\begin{matrix} (3.6) & J_{λ_{n}} (u) ⩽ J_{λ_{1}} (u) ⩽ 0, \forall u \in \hat{E} (u_{λ}) ∖ B_{R} (0) . \end{matrix}$ From (3.5), $J_{λ_{n}} (t_{n} u_{λ} + v_{n}) = {max}_{u \in \hat{E} (u_{λ})} J_{λ_{n}} ⩾ d_{λ_{n}} ⩾ d_{λ} > 0$ , namely $\begin{matrix} (3.7) & J_{λ_{n}} (t_{n} u_{λ} + v_{n}) > 0 . \end{matrix}$ Together with (3.6) and (3.7), we get that $‖ t_{n} u_{λ} + v_{n} ‖ ⩽ R$ , which implies that the sequence ${t_{n} u_{λ} + v_{n}}$ is bounded in $H^{α} (R^{N})$ , therefore we obtain that $\int_{R^{N}} F (t_{n} u_{λ} + v_{n}) d x$ is also bounded, $\begin{array}{l} d_{λ_{n}} & ⩽ J_{λ_{n}} (t_{n} u_{λ} + v_{n}) \\ = J_{λ} (t_{n} u_{λ} + v_{n}) + (λ - λ_{n}) \int_{R^{N}} F (t_{n} u_{λ} + v_{n}) d x \\ ⩽ J_{λ} (u_{λ}) + (λ - λ_{n}) \int_{R^{N}} F (t_{n} u_{λ} + v_{n}) d x \\ = d_{λ} + o_{n} (1), \end{array}$ together with $d_{λ} ⩽ d_{λ_{n}}$ for all $n \in N$ , we obtain ${lim}_{n \to + \infty} d_{λ_{n}} = d_{λ}$ .

The decreasing case: Choosing the sequence $λ_{1} ⩾ λ_{2} ⩾ \dots ⩾ λ_{n} \to λ$ , then $d_{λ_{1}} ⩽ d_{λ_{2}} ⩽ \dots ⩽ d_{λ_{n}} ⩽ d_{λ}$ . Let $u_{n}$ be the ground state solution of $(P_{λ_{n}})$ for each $n \in N$ , then there exist $t_{n} ⩾ 0$ and $v_{n} \in E^{-}$ such that $\begin{matrix} J_{λ} (t_{n} u_{n} + v_{n}) = max_{u \in \hat{E} (u_{n})} J_{λ} (u) . \end{matrix}$ Since $J_{λ}$ is coercive on $N_{λ}$ by Lemma 3.4, it is easy to prove that the sequence $(u_{n})$ is bounded.

Next we start to prove that ${lim}_{n \to + \infty} d_{λ_{n}} = d_{λ}$ . First, since $(u_{n})$ is bounded in $H^{α} (R^{N})$ , there exist $η > 0$ , $r > 0$ and $(y_{n}) \subset Z^{N}$ such that the following inequality is true for each $n \in N$ , $\begin{matrix} (3.8) & sup_{y_{n} \in Z^{N}} \int_{B_{r} (y_{n})} {| u_{n}^{+} |}^{2} d x ⩾ η . \end{matrix}$ If not, by Proposition 2.2, $u_{n}^{+} \to 0$ in $L^{p} (R^{N})$ for $p \in (2, 2_{α}^{*})$ . Then by $(f_{1})$ and $(f_{2})$ , we get $\int_{R^{N}} f (u_{n}) u_{n}^{+} d x \to 0$ , then $\begin{matrix} 0 = J_{λ_{n}}^{'} (u_{n}) u_{n}^{+} = {‖ u_{n}^{+} ‖}^{2} - λ_{n} \int_{R^{N}} f (u_{n}) u_{n}^{+} d x = {‖ u_{n}^{+} ‖}^{2} + o_{n} (1), \end{matrix}$ which means that $‖ u_{n}^{+} ‖^{2} \to 0$ , contradicting the equality $‖ u_{n}^{+} ‖^{2} ⩾ 2 d_{λ_{n}} > 0$ by Lemma 3.2. Thus (3.8) holds.

Let ${\tilde{u}}_{n} (x) : = u_{n} (x + y_{n})$ , we have that $({\tilde{u}}_{n})$ is bounded, then up to a subsequence, ${\tilde{u}}_{n}^{+} ⇀ {\tilde{u}}^{+} \neq 0$ in $H^{α} (R^{N})$ . Fixing $V : = {{\tilde{u}}_{n}^{+}}_{n \in N} \subset E^{+} ∖ {0}$ , then $V$ is bounded and the sequence $({\tilde{u}}_{n}^{+})$ does not weakly converge to zero in $H^{α} (R^{N})$ . Thus, by Lemma 3.3, there exists $R > 0$ such that for every $u \in V$ , $\begin{matrix} (3.9) & J_{λ} (ω) < 0 for ω \in E (u) ∖ B_{R} (0), \end{matrix}$ and let ${\tilde{v}}_{n} (x) : = v_{n} (x + y_{n})$ , we have $\begin{matrix} (3.10) & J_{λ} (t_{n} {\tilde{u}}_{n} + {\tilde{v}}_{n}) = J_{λ} (t_{n} u_{n} + v_{n}) = max_{u \in \hat{E} (u_{n})} J_{λ} (u) ⩾ d_{λ} > 0, \forall n \in N \end{matrix}$ Hence, by (3.9) and (3.10), we derive that $‖ t_{n} {\tilde{u}}_{n} + {\tilde{v}}_{n} ‖ ⩽ R$ for every $n \in N$ , then $‖ t_{n} u_{n} + v_{n} ‖ ⩽ R$ . Namely the sequence $(t_{n} u_{n} + v_{n})$ is bounded in $H^{α} (R^{N})$ , which implies that $\int_{R^{N}} F (t_{n} u_{n} + v_{n}) d x$ is bounded, thus, $\begin{matrix} d_{λ} & ⩽ J_{λ} (t_{n} u_{n} + v_{n}) = J_{λ_{n}} (t_{n} u_{n} + v_{n}) + (λ_{n} - λ) \int_{R^{N}} F (t_{n} u_{n} + v_{n}) d x \\ ⩽ d_{λ_{n}} + (λ_{n} - λ) \int_{R^{N}} F (t_{n} u_{n} + v_{n}) d x \\ = d_{λ_{n}} + o_{n} (1) . \end{matrix}$ This means that $\begin{matrix} d_{λ} ⩽ d_{λ_{n}} + o_{n} (1), \forall n \in N, \end{matrix}$ and together with $d_{λ} ⩾ d_{λ_{n}}$ for each $n \in N$ , we obtain ${lim}_{n \to + \infty} d_{λ_{n}} = d_{λ}$ . We complete the proof. □

4. Existence of ground state solution for equation ${(P)}_{ϵ}$

In this section, we start to prove the existence of ground state solutions for problem ( P ϵ ). Denote the energy functional associated with the problem ( P ϵ ) $\begin{matrix} I_{ϵ} (u) = \frac{1}{2} \int_{R^{N}} ({| {(- Δ)}^{\frac{α}{2}} u |}^{2} + V (x) | u |^{2}) d x - \int_{R^{N}} A (ϵ x) F (u) d x, \end{matrix}$ or the following equivalent form $\begin{matrix} I_{ϵ} (u) = \frac{1}{2} {‖ u^{+} ‖}^{2} - \frac{1}{2} {‖ u^{-} ‖}^{2} - \int_{R^{N}} A (ϵ x) F (u) d x, \end{matrix}$ and the corresponding Nehari manifold is $\begin{matrix} M_{ϵ} : = {u \in H^{α} (R^{N}) ∖ E^{-}; I_{ϵ}^{'} (u) u = 0 and I_{ϵ}^{'} (u) v = 0, \forall v \in E^{-}} . \end{matrix}$ Without loss of generality, we assume that $\begin{matrix} A (0) = sup_{x \in R^{N}} A (x) and A_{\infty} : = lim_{| x | \to + \infty} A (x) . \end{matrix}$

Next, we define the ground state energy of $I_{ϵ} (u)$ on Nehari manifold $c_{ϵ} = {inf}_{u \in M_{ϵ}} I_{ϵ} (u)$ . Moreover, the similar process in Lemma 3.5 can be used in $I_{ϵ} (u)$ , then for each $u \in H^{α} (R^{N}) ∖ E^{-}$ , there is only one point in $M_{ϵ} \cap \hat{E} (u)$ , hence, there exists a unique pair $t ⩾ 0$ and $v \in E^{-}$ such that $\begin{matrix} (4.1) & I_{ϵ} (t u + v) = max_{ω \in \hat{E} (u)} I_{ϵ} (ω) . \end{matrix}$ The ground state energy $c_{ϵ}$ of $I_{ϵ}$ can be given by $\begin{matrix} (4.2) & 0 < c_{ϵ} = inf_{u \in E^{+} ∖ {0}} max_{v \in \hat{E} (u)} I_{ϵ} (v) = inf_{u \in M_{ϵ}} I_{ϵ} (u) . \end{matrix}$ For convenience, we denote some notes $I_{0} = J_{A (0)}$ , $c_{0} = d_{A (0)}$ and $M_{0} = N_{A (0)}$ .

To begin with, we show a relation between $c_{0}$ and $c_{ϵ}$ in order to help us finish the below proofs.

Lemma 4.1.
${lim}_{ϵ \to 0} c_{ϵ} = c_{0}$ .
Proof.
Let $ϵ_{n} \to 0$ as $n \to + \infty$ , and $A (ϵ_{n} x) ⩽ A (0)$ for each $n \in N$ , thus $\begin{matrix} (4.3) & I_{0} (u) - I_{ϵ_{n}} (u) = \int_{R^{N}} (A (ϵ_{n} x) - A (0)) F (u) d x ⩽ 0, \forall u \in H^{α} (R^{N}) . \end{matrix}$ Let $c_{ϵ_{n}} = I_{ϵ_{n}} (u_{n})$ be the ground state energy of the functional $I_{ϵ_{n}} (u)$ for $u_{n} \in H^{α} (R^{N})$ . From the last section, there exists a unique pair $(m_{n}, z_{n})$ with $m_{n} \in [0, + \infty)$ and $z_{n} \in E^{-}$ such that $\begin{matrix} I_{0} (m_{n} u_{n}^{+} + z_{n}) = max_{u \in \hat{E} (u_{n})} I_{0} (u) . \end{matrix}$ By the definition of $c_{0}$ , we obtain $\begin{matrix} (4.4) & \begin{matrix} c_{0} & ⩽ I_{0} (m_{n} u_{n}^{+} + z_{n}) \\ = I_{ϵ_{n}} (m_{n} u_{n}^{+} + z_{n}) + \int_{R^{N}} (A (ϵ_{n} x) - A (0)) F (m_{n} u_{n}^{+} + z_{n}) d x \\ ⩽ c_{ϵ_{n}} + \int_{R^{N}} (A (ϵ_{n} x) - A (0)) F (m_{n} u_{n}^{+} + z_{n}) d x, \end{matrix} \end{matrix}$ hence, together with (4.3), we can deduce that $c_{0} ⩽ c_{ϵ_{n}}$ for each $n \in N$ and $c_{0} = {lim inf}_{n \to + \infty} c_{ϵ_{n}}$ .

Next, we know that the autonomous problem with $λ = A (0)$ has a ground state solution $u_{0}$ from Section 3, there exist $t_{n} \in [0, + \infty)$ , $v_{n} \in E^{-}$ such that the following equation holds, $\begin{matrix} max_{u \in \hat{E} (u_{0})} I_{n} (u) = I_{n} (t_{n} u_{0}^{+} + v_{n}) ⩾ c_{ϵ_{n}} ⩾ c_{0} > 0, \forall n \in N . \end{matrix}$ By Lemma 3.3, we know that the sequence ${t_{n} u_{0}^{+} + v_{n}}$ is bounded in $H^{α} (R^{N})$ . Then we assume that $t_{n} \to t_{0}$ and $v_{n} ⇀ v_{0}$ in $H^{α} (R^{N})$ and $c_{ϵ_{n}}$ satisfies $\begin{matrix} c_{ϵ_{n}} ⩽ I_{ϵ_{n}} (t_{n} u_{0}^{+} + v_{n}) = \frac{1}{2} t_{n}^{2} {‖ u_{0}^{+} ‖}^{2} - \frac{1}{2} ‖ v_{n} ‖^{2} - \int_{R^{N}} A (ϵ_{n} x) F (t_{n} u_{0}^{+} + v_{n}) d x . \end{matrix}$ Therefore, by the weakly lower semicontinuous of the norm, and the Fatou’s lemma, $\begin{array}{l} c_{0} & = \underset{n \to + \infty}{lim inf} c_{ϵ_{n}} ⩽ \underset{n \to + \infty}{lim sup} c_{ϵ_{n}} \\ ⩽ \underset{n \to + \infty}{lim sup} (\frac{1}{2} t_{n}^{2} {‖ u_{0}^{+} ‖}^{2} - \frac{1}{2} ‖ v_{n} ‖^{2} - \int_{R^{N}} A (ϵ_{n} x) F (t_{n} u_{0}^{+} + v_{n}) d x) \\ ⩽ \frac{1}{2} t_{0}^{2} {‖ u_{0}^{+} ‖}^{2} - \frac{1}{2} ‖ v ‖^{2} - \int_{R^{N}} A (0) F (t_{0} u_{0}^{+} + v) d x \\ = I_{0} (t_{0} u_{0}^{+} + v) ⩽ I_{0} (u_{0}) = c_{0}, \end{array}$ so we obtain ${lim}_{ϵ \to 0} c_{ϵ} = c_{0}$ . □
Remark 4.2.
From the last inequalities, we found that $I_{0} (t_{0} u_{0}^{+} + v) = I_{0} (u_{0}) = c_{0}$ , then both $t_{0} u_{0}^{+} + v$ and $u_{0}$ are the elements of $M_{0} \cap \hat{E} (u_{0})$ . But there is only one point in $M_{0} \cap \hat{E} (u_{0})$ , so $t_{0} u_{0}^{+} + v = u_{0}$ and $t_{n} \to t_{0} = 1$ , $v_{n} \to u_{0}^{-}$ .
Corollary 4.3.
There exists $ϵ_{0} > 0$ such that $c_{ϵ} < d_{A_{\infty}}$ for any $ϵ \in (0, ϵ_{0})$ .
Proof.
From assumption ( A ), we know $A (0) > A_{\infty}$ , then $d_{A_{\infty}} > c_{0}$ by Proposition 3.8. Because ${lim}_{ϵ \to 0} c_{ϵ} = c_{0}$ , choosing $ϵ_{0}$ small enough, we can obtain the result. □
Proposition 4.4.
The functional $I_{ϵ}$ is coercive on $M_{ϵ}$ for each $ϵ ⩾ 0$ .
Proof.
Suppose by contradiction, assume that there exists a sequence $(w_{n}) \subset M_{ϵ}$ satisfying $\begin{matrix} I_{ϵ} (w_{n}) ⩽ h as ‖ w_{n} ‖ \to + \infty, \end{matrix}$ where h is a fixed number in $(0, + \infty)$ . Let $v_{n} : = \frac{w_{n}}{‖ w_{n} ‖}$ , from Lemma 3.2, we know that $‖ v_{n}^{+} ‖ ⩾ ‖ v_{n}^{-} ‖$ , and then $‖ v_{n}^{+} ‖^{2} ⩾ \frac{1}{2}$ . So there exists a sequence $(y_{n}) \subset Z^{N}$ , $r > 0$ and $θ > 0$ satisfying the following inequality for each $n \in N$ , $\begin{matrix} (4.5) & sup_{y_{n} \in Z^{N}} \int_{B_{r} (y_{n})} {| v_{n}^{+} |}^{2} d x ⩾ θ . \end{matrix}$ If not, we can obtain $v_{n}^{+} \to 0$ in $L^{p} (R^{N})$ for all $p \in (2, 2_{α}^{})$ by Proposition 2.1, then by $(f_{1})$ and $(f_{2})$ , $\int_{R^{N}} A (ϵ x) F (m v_{n}^{+}) d x \to 0$ for each $m > 0$ . Hence, $\begin{array}{l} h & ⩾ I_{ϵ} (m v_{n}^{+}) = \frac{1}{2} m^{2} {‖ v_{n}^{+} ‖}^{2} - \int_{R^{N}} A (ϵ x) F (m v_{n}^{+}) d x \\ ⩾ \frac{1}{4} m^{2} - A (0) \int_{R^{N}} F (m v_{n}^{+}) d x \to \frac{1}{4} m^{2}, \end{array}$ because m is arbitrary but h has been fixed, it’s impossible, thus (4.5) holds.

Now we define ${\tilde{w}}_{n} (x) : = w_{n} (x + y_{n})$ and ${\tilde{v}}_{n} (x) : = v_{n} (x + y_{n})$ , we have ${\tilde{v}}_{n}^{+} (x) : = v_{n}^{+} (x + y_{n})$ and ${\tilde{w}}_{n} = {\tilde{v}}_{n} ‖ w_{n} ‖$ . Up to a subsequence if necessary, ${\tilde{v}}_{n}^{+} ⇀ {\tilde{v}}^{+} \neq 0$ in $H^{α} (R^{N})$ by (4.5). Hence ${\tilde{v}}_{n} (x) \to \tilde{v} (x) \neq 0$ a.e. in $R^{N}$ , and ${\tilde{w}}_{n} (x) = {\tilde{v}}_{n} (x) ‖ w_{n} ‖ \to + \infty$ a.e. in $R^{N}$ as $‖ w_{n} ‖ \to + \infty$ .

Then from the Fatou’s lemma and Lemma 3.2, $\begin{array}{l} \int_{R^{N}} \frac{F (w_{n})}{‖ w_{n} ‖^{2}} d x & = \int_{R^{N}} \frac{F (w_{n})}{| w_{n} |^{2}} | v_{n} |^{2} d x = \int_{R^{N}} \frac{F ({\tilde{w}}_{n})}{| {\tilde{w}}_{n} |^{2}}) | {\tilde{v}}_{n} |^{2} d x \\ ⩾ \int_{[v \neq 0]} \frac{F ({\tilde{w}}_{n})}{| {\tilde{w}}_{n} |^{2}} | {\tilde{v}}_{n} |^{2} d x \to + \infty, \end{array}$ where $[v \neq 0]$ is the Lebesgue measure of the set ${x \in R^{N} : v (x) \neq 0}$ .

Let $A_{\min} = {inf}_{x \in R^{N}} A (x)$ , then $\begin{array}{l} 0 & ⩽ \frac{I_{ϵ} (w_{n})}{‖ w_{n} ‖^{2}} = \frac{1}{2} {‖ v_{n}^{+} ‖}^{2} - \frac{1}{2} {‖ v_{n}^{-} ‖}^{2} - \int_{R^{N}} A (ϵ x) \frac{F (w_{n})}{‖ {\tilde{w}}_{n} ‖^{2}} d x \\ ⩽ \frac{1}{2} - A_{\min} \int_{R^{N}} \frac{F (w_{n})}{‖ {\tilde{w}}_{n} ‖^{2}} d x \to - \infty, \end{array}$ obtain a contradiction. The proof is completed. □
Theorem 4.5.
Under the assumptions* ( V ), ( A ) and $(f_{1})$ – $(f_{4})$ , the problem ( P ϵ ) has a ground state solution for each $ϵ \in (0, ϵ_{0})$ , where $ϵ_{0}$ was given by Corollary 4.3 .
Proof.
From Lemma 3.7 and using the Ekeland variational principle in [31], there exists a $(PS)$ sequence $(w_{n}) \subset M_{ϵ}$ for $c_{ϵ}$ such that $\begin{matrix} I_{ϵ} (w_{n}) \to c_{ϵ} and I_{ϵ}^{'} (w_{n}) \to 0 . \end{matrix}$ The sequence $(w_{n})$ is bounded in $H^{α} (R^{N})$ since $I_{ϵ}$ is coercive on $M_{ϵ}$ , we need to show that $w_{n} ⇀ w \neq 0$ in $H^{α} (R^{N})$ . By the definition of $(w_{n}) \subset M_{ϵ}$ , let $o_{n} (1)$ be a sequence converging to zero, we have $\begin{matrix} o_{n} (1) = I_{ϵ}^{'} (w_{n}) w_{n}^{+} = {‖ w_{n}^{+} ‖}^{2} - \int_{R^{N}} A (ϵ x) f (w_{n}) w_{n}^{+} d x ⩾ 2 c_{ϵ} - \int_{R^{N}} A (ϵ x) f (w_{n}) w_{n}^{+} d x . \end{matrix}$ Hence $\begin{matrix} \int_{R^{N}} A (ϵ x) f (w_{n}) w_{n}^{+} d x ⩾ 2 c_{ϵ} > 0, \end{matrix}$ it follows that $\int_{R^{N}} f (w_{n}) w_{n}^{+} d x > 0$ , and $(w_{n})$ is bounded in $H^{α} (R^{N})$ , then from Proposition 2.2, there exists a sequence $(y_{n}) \subset Z^{N}$ , $r > 0$ and $μ > 0$ such that $\begin{matrix} (4.6) & \int_{B_{r} (y_{n})} {| w_{n}^{+} |}^{2} d x ⩾ μ > 0, \forall n \in N . \end{matrix}$ First, we prove that the sequence $(y_{n})$ is bounded in $R^{N}$ , suppose by contradiction, assume that $(y_{n})$ is unbounded and $| y_{n} | \to + \infty$ as $n \to \infty$ . Define $v_{n} (x) : = w_{n} (x + y_{n})$ , then $v_{n} ⇀ v \neq 0$ by (4.6). Choosing the test function $ϕ \in C_{0}^{\infty} (R^{N})$ , we obtain $\begin{matrix} (4.7) & \begin{matrix} o_{n} (1) & = I_{ϵ}^{'} (w_{n}) ϕ (\cdot - y_{n}) \\ = \int_{R^{N}} {(- Δ)}^{α} w_{n} ϕ (\cdot - y_{n}) + V (x) w_{n} ϕ (\cdot - y_{n}) d x \\ - \int_{R^{N}} A (ϵ x) f (w_{n}) ϕ (\cdot - y_{n}) d x \\ = \int_{R^{N}} {(- Δ)}^{α} v_{n} ϕ + V (x) v_{n} ϕ d x - \int_{R^{N}} A (ϵ x + ϵ y_{n}) f (v_{n}) ϕ d x . \end{matrix} \end{matrix}$ Passing to the limit $n \to + \infty$ , we get $\begin{matrix} \int_{R^{N}} {(- Δ)}^{α} v ϕ + V (x) v ϕ d x - \int_{R^{N}} A_{\infty} f (v) ϕ d x = J_{A_{\infty}}^{'} (v) ϕ = 0, \forall ϕ \in C_{0}^{\infty} (R^{N}) . \end{matrix}$ By the density of $C_{0}^{\infty} (R^{N})$ in $H^{α} (R^{N})$ , we obtained $\begin{matrix} (4.8) & \int_{R^{N}} {(- Δ)}^{α} v ψ + V (x) v ψ d x - \int_{R^{N}} A_{\infty} f (v) ψ d x = J_{A_{\infty}}^{'} (v) ψ = 0, \forall ψ \in H^{α} (R^{N}) . \end{matrix}$ This implies that v is a nontrival solution of $(P_{A_{\infty}})$ and $v \in N_{A_{\infty}}$ , by the Fatou’s lemma, $\begin{matrix} (4.9) & \begin{matrix} d_{A_{\infty}} & ⩽ J_{A_{\infty}} (v) = J_{A_{\infty}} (v) - \frac{1}{2} J_{A_{\infty}}^{'} (v) v = \int_{R^{N}} A_{\infty} (\frac{1}{2} f (v) v - F (v)) d x \\ ⩽ \underset{n \to + \infty}{lim inf} \int_{R^{N}} A (ϵ x + ϵ y_{n}) (\frac{1}{2} f (v_{n}) v_{n} - F (v_{n})) d x \\ = \underset{n \to + \infty}{lim inf} \int_{R^{N}} A (ϵ x) (\frac{1}{2} f (w_{n}) w_{n} - F (w_{n})) d x \\ = \underset{n \to + \infty}{lim inf} (I_{ϵ} (w_{n}) - \frac{1}{2} I_{ϵ}^{'} (w_{n}) w_{n}) = c_{ϵ} . \end{matrix} \end{matrix}$ It follows that $d_{A_{\infty}} ⩽ c_{ϵ}$ , this yields a contradiction since $c_{ϵ} < d_{A_{\infty}}$ for any $ϵ \in (0, ϵ_{0})$ by Corollary 4.3. Hence $(y_{n})$ is bounded, we can select $R > 0$ such that $B_{r} (y_{n}) \subset B_{R} (0)$ for all $n \in N$ and $\begin{matrix} \int_{B_{R} (0)} {| w_{n}^{+} |}^{2} d x ⩾ \int_{B_{r} (y_{n})} {| w_{n}^{+} |}^{2} d x ⩾ μ, \forall n \in N, \end{matrix}$ we deduce that $w_{n} ⇀ w \neq 0$ . Repeating the method in (4.7) and (4.8), we can obtain that $w \in M_{ϵ}$ is a nontrival solution for ( P ϵ ), thus $c_{ϵ} ⩽ I_{ϵ} (w)$ .

On the other hand, by the Fatou’s lemma and $(f_{4})$ , we have $\begin{array}{l} c_{ϵ} & = \underset{n \to + \infty}{lim inf} (I_{ϵ} (w_{n}) - \frac{1}{2} I_{ϵ}^{'} (w_{n}) w_{n}) \\ = \underset{n \to + \infty}{lim inf} \int_{R^{N}} A (ϵ x) (\frac{1}{2} f (w_{n}) w_{n} - F (w_{n})) d x \\ ⩾ \int_{R^{N}} A (ϵ x) (\frac{1}{2} f (w) w - F (w)) d x \\ = I_{ϵ} (w) - \frac{1}{2} I_{ϵ}^{'} (w) w = I_{ϵ} (w) . \end{array}$ Hence $c_{ϵ} = I_{ϵ} (w)$ , it follows that w is a ground state solution of ( P ϵ ). We complete the proof. □

5. The concentration result of ground state solutions

In this section, we start to prove the concentration result of the maximum points of the solutions which was given by the last section. For convenience, we denote $I_{ϵ_{n}} = I_{n}$ , $c_{ϵ_{n}} = c_{n} = I_{ϵ_{n}} (u_{ϵ_{n}})$ , $u_{ϵ_{n}} = u_{n}$ , and $I_{n}^{'} (u_{n}) = 0$ , where $(ϵ_{n})$ is a sequence in $(0, ϵ_{0})$ such that $ϵ_{n} \to 0$ as $n \to + \infty$ . From the argument in Lemma 4.1, we have $0 < c_{0} ⩽ c_{n}$ for all $n \in N$ . Define $A = {x \in R^{N} : A (x) = A (0)}$ is the set of the maximum points of $A (x)$ . We need to show that if $x_{n}$ is the maximum point of $| u_{n} |$ for every $n \in N$ , then $\begin{matrix} lim_{ϵ_{n} \to 0} A (ϵ_{n} x_{n}) = A (z), \end{matrix}$ where $z \in A$ . In other words, we need to prove that for the sequence $ϵ_{n} \to 0$ as $n \to \infty$ , up to a subsequence if necessary, $ϵ_{n} x_{n} \to z \in A$ .

The following lemmas are important tools to prove the main result of this section.

Lemma 5.1.
The sequence $(u_{n})$ is bounded in $H^{α} (R^{N})$ .
Proof.
It is easy to check this point by Proposition 4.4. □
Lemma 5.2.
There exist $(y_{n}) \subset Z^{N}$ , $R > 0$ and $η > 0$ such that $\begin{matrix} \int_{B_{R} (y_{n})} {| u_{n}^{+} |}^{2} d x ⩾ η . \end{matrix}$ Furthermore, the sequence $(ϵ_{n} y_{n})$ is bounded in $R^{N}$ and ${lim}_{n \to + \infty} ϵ_{n} y_{n} = z$ , $z \in A$ .
Proof.
First of all, since $(u_{n})$ is bounded, we want to show that there exist $(y_{n}) \subset Z^{N}$ , $R > 0$ and $η > 0$ such that $\begin{matrix} \int_{B_{R} (y_{n})} {| u_{n}^{+} |}^{2} d x ⩾ η . \end{matrix}$ Suppose by contradiction, we obtain that $u_{n}^{+} \to 0$ in $L^{p} (R^{N})$ for $p \in (2, 2_{α}^{})$ by Proposition 2.2. By ( A ), $(f_{1})$ and $(f_{2})$ , $\begin{matrix} 0 ⩽ \int_{R^{N}} A (ϵ_{n} x) f (u_{n}) u_{n}^{+} d x ⩽ \int_{R^{N}} A (0) f (u_{n}) u_{n}^{+} d x \to 0, \end{matrix}$ therefore, $‖ u_{n}^{+} ‖^{2} = \int_{R^{N}} A (ϵ_{n} x) f (u_{n}) u_{n}^{+} d x \to 0$ , which is impossible because $‖ u_{n}^{+} ‖ ⩾ \sqrt{2 c_{n}} ⩾ \sqrt{2 c_{0}} > 0$ by Lemma 3.2. Define the sequence $v_{n} (x) : = u_{n} (x + y_{n})$ , then for some subsequence $v_{n} ⇀ v \neq 0$ .

Next, we prove the sequence $(ϵ_{n} y_{n})$ is bounded in $R^{N}$ by contradiction. Suppose there exists a subsequence such that $| ϵ_{n} y_{n} | \to + \infty$ as $n \to \infty$ , then we use the same idea in the proof of Theorem 4.5. Define $u_{n}$ is a ground state solution of $(P_{ϵ_{n}})$ , for any $ϕ \in C_{0}^{\infty} (R^{N})$ , we have $\begin{array}{l} 0 & = I_{n}^{'} (u_{n}) ϕ (\cdot - y_{n}) \\ = \int_{R^{N}} {(- Δ)}^{α} u_{n} ϕ (\cdot - y_{n}) + V (x) u_{n} ϕ (\cdot - y_{n}) d x - \int_{R^{N}} A (ϵ_{n} x) f (u_{n}) ϕ (\cdot - y_{n}) d x \\ = \int_{R^{N}} {(- Δ)}^{α} v_{n} ϕ + V (x) v_{n} ϕ d x - \int_{R^{N}} A (ϵ_{n} x + ϵ_{n} y_{n}) f (v_{n}) ϕ d x . \end{array}$ Passing the limit $n \to + \infty$ , and by the density of $C_{0}^{\infty} (R^{N})$ in $H^{α} (R^{N})$ , we have $\begin{matrix} \int_{R^{N}} {(- Δ)}^{α} v φ + V (x) v φ d x - \int_{R^{N}} A_{\infty} f (v) φ d x = 0, \forall φ \in H^{α} (R^{N}) . \end{matrix}$ It implies that v is a nontrivial solution of $(P_{A_{\infty}})$ and $v \in M_{\infty}$ . By the Fatou’s lemma and Lemma 4.1, $\begin{array}{l} d_{A_{\infty}} & ⩽ J_{A_{\infty}} (v) = J_{A_{\infty}} (v) - \frac{1}{2} J_{A_{\infty}}^{'} (v) = \int_{R^{N}} A_{\infty} (\frac{1}{2} f (v) v - F (v)) d x \\ ⩽ \underset{n \to + \infty}{lim inf} \int_{R^{N}} A (ϵ_{n} x + ϵ_{n} y_{n}) (\frac{1}{2} f (v_{n}) v_{n} - F (v_{n})) d x \\ = \underset{n \to + \infty}{lim inf} \int_{R^{N}} A (ϵ_{n} x) (\frac{1}{2} f (u_{n}) u_{n} - F (u_{n})) d x \\ = \underset{n \to + \infty}{lim inf} (I_{n} (u_{n}) - \frac{1}{2} I_{n}^{'} (u_{n}) u_{n}) \\ = \underset{n \to + \infty}{lim inf} I_{n} (u_{n}) = lim_{n \to + \infty} c_{ϵ_{n}} = c_{0} . \end{array}$ Together with $c_{0} < d_{A_{\infty}}$ by Proposition 3.8, we get a contradiction. So the sequence $(ϵ_{n} y_{n})$ is bounded in $R^{N}$ . Then up to a subsequence, suppose that $ϵ_{n} y_{n} \to z$ . Repeating above process, we have $\begin{matrix} (5.1) & \int_{R^{N}} {(- Δ)}^{α} v φ + V (x) v φ d x - \int_{R^{N}} A (z) f (v) φ d x = 0, \forall φ \in H^{α} (R^{N}) . \end{matrix}$ It follows that v is a nontrivial solution of equation $(P_{A (z)})$ and $v \in M_{A (z)}$ . Similarly, by the above method, we obtained $d_{A (z)} ⩽ c_{0} = d_{A (0)}$ . Hence $A (z) ⩾ A (0)$ by Lemma 3.8, while according to condition ( A ), we deduce that $A (z) = A (0)$ . So we obtain ${lim}_{n \to + \infty} ϵ_{n} y_{n} = z$ and $z \in A$ . □
Lemma 5.3.
For each* $η > 0$ , there exists $σ : = σ_{η} \in (0, 1)$ and $c_{η} > 0$ such that $\begin{matrix} \frac{| f (t) |}{| t |} ⩽ η, \forall t \in [σ, σ], \end{matrix}$ and $\begin{matrix} {| f (t) |}^{\frac{q + 1}{q}} ⩽ c_{η} t f (t), \forall t \in R ∖ [- σ, σ], \end{matrix}$ where $q \in (1, 2_{α}^{} - 1)$ .
Proof.
Because $f (t)$ is continuous in $R$ and $\frac{| f (t) |}{| t |} \to 0$ as $| t | \to 0$ by condition $(f_{1})$ , hence $\frac{| f (t) |}{| t |}$ is bounded in $[- σ, σ]$ , the first inequality holds for some η.

We consider the second inequality in two intervals $H : = [- 1, - σ] \cup [σ, 1]$ and $G : = (- \infty, - 1] \cup [1, + \infty)$ .

Because H is a bounded closed subset, by the continuity of f, so $f (t)$ is bounded in H, then there exists a number $η_{1} > 0$ such that $| f (t) |^{\frac{1}{q}} ⩽ η_{1}$ and $| f (t) |^{\frac{q + 1}{q}} ⩽ η_{1} t f (t)$ for all $t \in H$ .

From conditions $(f_{1})$ , $(f_{2})$ , we can get $\begin{matrix} | f (t) | ⩽ c_{1} | t | + c_{2} | t |^{q}, \end{matrix}$ where $c_{1}$ , $c_{2}$ are some fixed constants. Because $q > 1$ and $| t | ⩾ 1$ , it’s easy to see that $\begin{matrix} {| f (t) |}^{\frac{1}{q}} ⩽ {(c_{1} | t | + c_{2} | t |^{q})}^{\frac{1}{q}} ⩽ c^{\frac{1}{q}} | t |^{\frac{1}{q}} ⩽ c^{\frac{1}{q}} | t |, \forall t \in G, \end{matrix}$ where $c = max {c_{1}, c_{2}}$ .

Multiplying $| f (t) |$ on both sides, we get $\begin{matrix} {| f (t) |}^{\frac{q + 1}{q}} ⩽ c^{\frac{1}{q}} | t | | f (t) | = c^{\frac{1}{q}} t f (t), \forall t \in G . \end{matrix}$ Based on the above argument, we can get the desired conclusion. □
Lemma 5.4.
The sequence* $(v_{n})$ strongly converges to v in $H^{α} (R^{N})$ .
Proof.
First, (5.1) shows that $v \in M_{0}$ . Using the method in Lemma 5.2 again, we obtain $\begin{array}{l} c_{0} & ⩽ I_{0} (v) = I_{0} (v) - \frac{1}{2} I_{0}^{'} (v) v \\ = \int_{R^{N}} A (0) (\frac{1}{2} f (v) v - F (v)) d x \\ = \int_{R^{N}} A (z) (\frac{1}{2} f (v) v - F (v)) d x \\ ⩽ \underset{n \to + \infty}{lim inf} \int_{R^{N}} A (ϵ_{n} x + ϵ_{n} y_{n}) (\frac{1}{2} f (v_{n}) v_{n} - F (v_{n})) d x \\ ⩽ \underset{n \to + \infty}{lim sup} \int_{R^{N}} A (ϵ_{n} x + ϵ_{n} y_{n}) (\frac{1}{2} f (v_{n}) v_{n} - F (v_{n})) d x \\ = \underset{n \to + \infty}{lim sup} \int_{R^{N}} A (ϵ_{n} x) (\frac{1}{2} f (u_{n}) u_{n} - F (u_{n})) d x \\ = \underset{n \to + \infty}{lim sup} (I_{n} (u_{n}) - \frac{1}{2} I_{n}^{'} (u_{n}) u_{n}) = lim_{n \to + \infty} c_{n} = c_{0} . \end{array}$ It implies that v is also a ground state solution of $(P_{A (0)})$ , and $\begin{matrix} lim_{n \to + \infty} \int_{R^{N}} A (ϵ_{n} x + ϵ_{n} y_{n}) (\frac{1}{2} f (v_{n}) v_{n} - F (v_{n})) d x = \int_{R^{N}} A (z) (\frac{1}{2} f (v) v - F (v)) d x . \end{matrix}$ Therefore, $\begin{matrix} \int_{R^{N}} A (ϵ_{n} x + ϵ_{n} y_{n}) (\frac{1}{2} f (v_{n}) v_{n} - F (v_{n})) d x \to \int_{R^{N}} A (z) (\frac{1}{2} f (v) v - F (v)) d x in L^{1} (R^{N}) . \end{matrix}$ From $(f_{4})$ , we obtain that $\begin{matrix} \int_{R^{N}} A (ϵ_{n} x + ϵ_{n} y_{n}) (\frac{1}{2} f (v_{n}) v_{n} - F (v_{n})) d x ⩾ 0, \forall n \in N . \end{matrix}$ From Lemma 5.1, the sequence $(v_{n})$ is bounded in $H^{α} (R^{N})$ , then up to a subsequence $v_{n} (x) ⇀ v (x)$ in $H^{α} (R^{N})$ , then $v_{n} \to v$ in $L_{loc}^{q} (R^{N})$ with $q \in (2, 2_{α}^{})$ , and $v_{n} \to v$ a.e. in $R^{N}$ . For some subsequence, and from [8, Theorem 4.9], there exists $T (x) \in L^{1} (R^{N})$ such that $\begin{matrix} 0 < A (ϵ_{n} x + ϵ_{n} y_{n}) (\frac{1}{2} f (v_{n}) v_{n} - F (v_{n})) ⩽ A (0) (\frac{1}{2} f (v_{n}) v_{n} - F (v_{n})) ⩽ T (x) a.e. in R^{N} . \end{matrix}$ Using $(f_{4})$ , there exists $μ > 2$ such that $\begin{matrix} 0 < A (0) (\frac{1}{2} - \frac{1}{μ}) f (v_{n}) v_{n} ⩽ T (x), \forall n \in N . \end{matrix}$ Therefore, there is a constant $c > 0$ such that $f (v_{n}) v_{n} ⩽ c T (x)$ for each $n \in N$ .

Next we define the function $\begin{matrix} D_{n} : = f (v_{n}) v_{n}^{+} - f (v) v^{+} . \end{matrix}$ We want to show $\int_{R^{N}} | D_{n} | d x \to 0$ .

From $v_{n} \to v$ in $L_{loc}^{q} (R^{N})$ with $q \in (2, 2_{α}^{})$ and $(f_{2})$ , we deduce that $f (v_{n}) v_{n}^{+} \to f (v) v^{+}$ in $B_{R} (0)$ for any $R > 0$ , $\begin{matrix} (5.2) & \int_{B_{R} (0)} | D_{n} | d x \to 0, \forall n \in N . \end{matrix}$ By Lemma 5.3, there exist positive numbers σ and η such that $\begin{matrix} \frac{| f (t) |}{| t |} ⩽ η, \forall t \in [- σ, σ], \end{matrix}$ and $\begin{matrix} {| f (t) |}^{\frac{q + 1}{q}} ⩽ c_{η} t f (t), \forall t \in R ∖ [- σ, σ] . \end{matrix}$ Let $[v (x) ⩽ - σ]$ and $[v (x) ⩾ σ]$ be the measures of ${x \in R^{N} : v (x) ⩽ - σ}$ and ${x \in R^{N} : v (x) ⩾ σ}$ , respectively. Define $W : = {sup}_{n \in N} {{(\int | v_{n}^{+} |^{q + 1} d x)}^{\frac{1}{q + 1}}, \int | v_{n} v_{n}^{+} | d x}$ . From the Hölder inequality, we have $\begin{array}{l} \int_{B_{R} {(0)}^{c}} | f (v_{n}) v_{n}^{+} | d x & = \int_{B_{R} {(0)}^{c} \cap [v (x) ⩽ σ]} | f (v_{n}) v_{n}^{+} | d x + \int_{B_{R} {(0)}^{c} \cap [v (x) ⩾ σ]} | f (v_{n}) v_{n}^{+} | d x \\ ⩽ \int_{B_{R} {(0)}^{c} \cap [v (x) ⩽ σ]} η | v_{n} v_{n}^{+} | d x \\ + {(\int_{B_{R} {(0)}^{c} \cap [v (x) ⩾ σ]} {| f (v_{n}) |}^{\frac{q + 1}{q}} d x)}^{\frac{q}{q + 1}} {(\int_{B_{R} {(0)}^{c} \cap [v (x) ⩾ σ]} {| v_{n}^{+} |}^{q + 1} d x)}^{\frac{1}{q + 1}} \\ ⩽ η W + c_{η} W {(\int_{B_{R} {(0)}^{c} \cap [v (x) ⩾ σ]} | f (v_{n}) v_{n} | d x)}^{\frac{1}{q + 1}} . \end{array}$ Let R large enough if necessary, such that the following inequality holds for each $n \in N$ , $\begin{matrix} c_{η} {(\int_{B_{R} {(0)}^{c} \cap [v (x) ⩾ σ]} | f (v_{n}) v_{n} | d x)}^{\frac{1}{q + 1}} ⩽ η . \end{matrix}$ Therefore, $\begin{matrix} \int_{B_{R} {(0)}^{c}} | f (v_{n}) v_{n}^{+} | d x ⩽ 2 η W, \forall n \in N . \end{matrix}$ Together with (5.2), it follows that $\begin{matrix} \int_{R^{N}} | D_{n} | d x \to 0, \end{matrix}$ and $\begin{matrix} f (v_{n}) v_{n}^{+} \to f (v) v^{+} in L^{1} (R^{N}) . \end{matrix}$ Using the similar method, we can prove that $\begin{matrix} f (v_{n}) v_{n}^{-} \to f (v) v^{-} in L^{1} (R^{N}) . \end{matrix}$ Hence, we have $\begin{matrix} \int_{R^{N}} A (ϵ_{n} x + ϵ_{n} y_{n}) f (v_{n}) v_{n}^{+} d x \to \int_{R^{N}} A (0) f (v) v^{+} d x . \end{matrix}$ Since $I_{n}^{'} (v_{n}) v_{n}^{+} = 0$ and $I_{0}^{'} (v) v^{+} = 0$ , we obtain $\begin{matrix} {‖ v_{n}^{+} ‖}^{2} = \int_{R^{N}} A (ϵ_{n} x + ϵ_{n} y_{n}) f (v_{n}) v_{n}^{+} d x and {‖ v^{+} ‖}^{2} = \int_{R^{N}} A (0) f (v) v^{+} d x . \end{matrix}$ It means that $v_{n}^{+} \to v^{+}$ in $H^{α} (R^{N})$ for some subsequence. Likewise, we have $v_{n}^{-} \to v^{-}$ in $H^{α} (R^{N})$ . Then from $v_{n} = v_{n}^{+} + v_{n}^{-}$ to know that $v_{n} \to v$ in $H^{α} (R^{N})$ . We finish the proof. □
Corollary 5.5.
$‖ v_{n} ‖_{L^{\infty} (R^{N})} ↛ 0$ .
Proof.
Suppose by contradiction, from Lemma 5.4, we get $v = 0$ , which is a contradiction with $v \in M_{A (0)}$ . □

To get the $L^{\infty}$ -estimate of the solution sequence $(v_{n})$ , we first introduce the Bessel kernel to show the solution of the following equation $\begin{matrix} (5.3) & {(- Δ)}^{α} v_{n} + v_{n} = h_{n} (x, v_{n}) in R^{N}, \end{matrix}$ where $h_{n} (x, v_{n}) = A (ϵ_{n} x + ϵ_{n} y_{n}) f (v_{n}) - (V (x) - 1) v_{n}$ .

Using the result in [13], the solution of (5.3) can be expressed as $\begin{matrix} (5.4) & v_{n} (x) = K * h_{n} (y, v_{n}) = \int_{R^{N}} K (x - y) h_{n} (y, v_{n}) d y, \end{matrix}$ where $K$ is the Bessel kernel.

Next, we introduce some properties of the Bessel kernel that will be used below, more details can be found in [13,29].
Lemma 5.6 ([13]).

$K$ is positive, radially symmetric and smooth in $R^{N} ∖ {0}$ .

there exist some constants $C_{1}$ , $C_{2}$ such that $\begin{matrix} K (x) ⩽ \frac{C_{1}}{| x |^{N + 2 α}} if | x | ⩾ 1 and K (x) ⩽ \frac{C_{2}}{| x |^{N - 2 α}} if | x | ⩽ 1 . \end{matrix}$

$K \in L^{p} (R^{N})$ for all $p \in [1, \frac{N}{N - 2 α})$ .

Lemma 5.7.

Let $v_{n}$ be a solution of $(P_{ϵ_{n}})$ , then there exists $p_{n} > \frac{N}{2 α}$ such that $v_{n} \in L^{p} (R^{N})$ , $p \in [2, p_{n}]$ for each $n \in N$ , and $v_{n} \to v$ in $L^{m} (R^{N})$ for $m \in [2, p_{n})$ .

Furthermore, there exists $q_{0}$ verifying $\frac{N}{2 α} < q_{0} < p_{n}$ such that $v_{n} \in L^{q_{0}} (R^{N})$ and $v_{n} \to v$ in $L^{q_{0}} (R^{N})$ for all $n \in N$ .

Proof.

From [13, Theorem 3.4], we can directly obtain that there exists $p_{n} > \frac{N}{2 α}$ such that $v_{n} \in L^{p_{n}} (R^{N})$ for each $n \in N$ . By Proposition 2.1, we know that $v_{n} \in L^{p} (R^{N})$ for $p \in [2, 2_{α}^{*}]$ . For each $m \in (2_{α}^{*}, p_{n})$ , there exists $t_{m} \in (0, 1)$ such that $\begin{matrix} \frac{1}{m} = \frac{t_{m}}{2_{α}^{*}} + \frac{1 - t_{m}}{p_{n}} . \end{matrix}$ Then using the Hölder inequalities, it follows that $v_{n} \in L^{m} (R^{N})$ for $m \in (2_{α}^{*}, p_{n})$ and for every $n \in N$ .

From Lemma 5.1 and Lemma 5.4, $v_{n} \to v$ in $H^{α} (R^{N})$ , hence $v_{n} \to v$ in $L^{2} (R^{N})$ by Sobolev embedding. For each $s \in (2, p_{n})$ , there exists $w_{s} \in (0, 1)$ such that $\begin{matrix} \frac{1}{s} = \frac{w_{s}}{2} + \frac{1 - w_{s}}{p_{n}} . \end{matrix}$ Define $ω_{n} : = v_{n} - v$ , using the Hölder inequalities again, $\begin{matrix} (5.5) & ‖ ω_{n} ‖_{L^{s} (R^{N})}^{s} ⩽ ‖ ω_{n} ‖_{L^{2} (R^{N})}^{w_{s}} ‖ ω_{n} ‖_{L^{p_{n}} (R^{N})}^{1 - w_{s}} \to 0 as n \to + \infty, \end{matrix}$ as $ω_{n} \to 0$ in $L^{2} (R^{N})$ and $ω_{n} \in L^{p_{n}} (R^{N})$ , $v_{n} \to v$ in $L^{m} (R^{N})$ for $m \in [2, p_{n})$ . We complete the proof. □

Lemma 5.8.

The sequence $(v_{n})$ satisfies $| v_{n} (x) | \to 0$ as $| x | \to \infty$ for all $n \in N$ .

Proof.

First, for each $σ \in (0, 1)$ , we define two sets $\begin{matrix} A_{σ} : = A_{σ} (x) = {y \in R^{N} : | x - y | ⩾ \frac{1}{σ}}, \end{matrix}$ and $\begin{matrix} B_{σ} : = B_{σ} (x) = {y \in R^{N} : | x - y | ⩽ \frac{1}{σ}} . \end{matrix}$ From (5.4), $\begin{matrix} | v_{n} (x) | ⩽ \int_{R^{N}} K (x - y) | h_{n} (y, v_{n}) | d y = \int_{A_{σ}} K (x - y) | h_{n} (y, v_{n}) | d y + \int_{B_{σ}} K (x - y) | h_{n} (y, v_{n}) | d y . \end{matrix}$ From Lemma 5.1, $(v_{n})$ is bounded in $H^{α} (R^{N})$ , by the continuous embedding in Proposition 2.1, $‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})} ⩽ C ‖ v_{n} ‖_{H^{α} (R^{N})}$ , which implies that $(v_{n})$ is bounded in $L^{2_{α}^{*}} (R^{N})$ .

Because $A (x)$ , $V (x)$ are bounded in $R^{N}$ , and $| f (u) | ⩽ C_{1} | u |^{p}$ , where $p \in (1, \frac{N + 2 α}{N - 2 α})$ by $(f_{2})$ , we have $\begin{matrix} (5.6) & | h_{n} (x, u_{n}) | ⩽ C_{2} (| u_{n} | + | u_{n} |^{p}), \end{matrix}$ where $C_{1}$ , $C_{2}$ are some constants.

Define $e = 2_{α}^{*}$ , $s_{0} = \frac{e}{e - 1}$ , $s_{1} = \frac{e}{e - p}$ , then $\begin{matrix} \frac{1}{s_{0}} + \frac{1}{e} = 1 and \frac{1}{s_{1}} + \frac{p}{e} = 1, \end{matrix}$ where p is from (5.6). It is easy to check $\begin{matrix} (5.7) & s_{0} > s_{1} > 1 and (N + 2 α) s_{0} > (N + 2 α) s_{1} > N + 2 α . \end{matrix}$ From (5.6), Lemma 5.6(b) and the Hölder inequalities, we have the following estimate for all $n \in N$ , $\begin{array}{l} \int_{A_{σ}} K (x - y) | h_{n} (y, v_{n}) | d y \\ ⩽ C_{2} \int_{A_{σ}} K (x - y) (| v_{n} (y) | + {| v_{n} (y) |}^{p}) d y \\ ⩽ C_{2} {(\int_{A_{σ}} K {(x - y)}^{s_{0}} d y)}^{\frac{1}{s_{0}}} ‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})} + C_{2} {(\int_{A_{σ}} K {(x - y)}^{s_{1}} d y)}^{\frac{1}{s_{1}}} ‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})}^{p} \\ ⩽ C_{2} {(\int_{A_{σ}} \frac{1}{| x - y |^{s_{0} (N + 2 α)}} d y)}^{\frac{1}{s_{0}}} ‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})} + C_{2} {(\int_{A_{σ}} \frac{1}{| x - y |^{s_{1} (N + 2 α)}} d y)}^{\frac{1}{s_{1}}} ‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})}^{p} \\ ⩽ C_{2} {(\int_{A_{σ}} \frac{1}{| x - y |^{N + 2 α}} d y)}^{\frac{1}{s_{0}}} ‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})} + C_{2} {(\int_{A_{σ}} \frac{1}{| x - y |^{N + 2 α}} d y)}^{\frac{1}{s_{1}}} ‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})}^{p} \\ ⩽ C_{2} (‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})} + ‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})}^{p}) {(\int_{A_{σ}} \frac{1}{| x - y |^{N + 2 α}} d y)}^{\frac{1}{s_{1}}} \\ ⩽ C_{2} σ^{t} (‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})} + ‖ v_{n} ‖_{L^{2_{α}^{*}} (R^{N})}^{p}) {(\int_{A_{σ}} \frac{1}{| x - y |^{N + α}} d y)}^{\frac{1}{s_{1}}} \\ ⩽ P σ^{t}, \end{array}$ where $t = \frac{α}{s_{1}}$ and P is a constant.

Next, we start to estimate the second term. From Lemma 5.7, there exists $q_{0} > \frac{N}{2 α}$ such that $v_{n} \in L^{q_{0}} (R^{N})$ and $v_{n} \to v \in L^{q_{0}} (R^{N})$ for each $n \in N$ . We define $\begin{matrix} \bar{q_{1}} = \frac{q_{0}}{q_{0} - 1} and \bar{q_{2}} = \frac{q_{0}}{q_{0} - p} . \end{matrix}$

We can check that $1 < \bar{q_{2}} < \bar{q_{1}} < \frac{N}{N - 2 α}$ holds. From Lemma 5.6(c), we have $K \in L^{{\bar{q}}_{1}} (R^{N})$ and $K \in L^{{\bar{q}}_{2}} (R^{N})$ , combining with the Hölder inequality, we obtain $\begin{array}{l} \int_{B_{σ}} K (x - y) | h_{n} (y, v_{n}) | d y & ⩽ C_{2} \int_{B_{σ}} K (x - y) (| v_{n} (y) | + {| v_{n} (y) |}^{p}) d y \\ ⩽ C_{2} ‖ K ‖_{L^{\bar{q_{1}}} (R^{N})} ‖ v_{n} ‖_{L^{q_{0}} (B_{σ})} + C_{2} ‖ K ‖_{L^{\bar{q_{2}}} (R^{N})} ‖ v_{n} ‖_{L^{q_{0}} (B_{σ})}^{p} \\ ⩽ C (‖ v_{n} ‖_{L^{q_{0}} (B_{σ})} + ‖ v_{n} ‖_{L^{q_{0}} (B_{σ})}^{p}) . \end{array}$ From Lemma 5.4, we can choose $R_{σ} > 0$ large enough for each σ such that the following inequality holds for all $n \in N$ , $\begin{matrix} {‖ v_{n} (x) ‖}_{L^{q_{0}} (B_{σ})} ⩽ σ, \forall | x | ⩾ R_{σ} . \end{matrix}$ Hence, due to $σ < 1$ , we have $\begin{matrix} (5.8) & \int_{B_{σ}} K (x - y) | h_{n} (y, v_{n}) | d y ⩽ C σ . \end{matrix}$ Then $\begin{matrix} (5.9) & | v_{n} (x) | ⩽ \int_{R^{N}} K (x - y) | h_{n} (y, v_{n}) | d y ⩽ P σ^{t} + C σ ⩽ Q σ, \forall | x | ⩾ R, \end{matrix}$ where Q is a constant related to P and C. For each σ, there exists $R ⩾ R_{σ}$ for each σ such that $v_{n} (x) ⩽ σ$ as $x \in R^{N} ∖ B_{R} (0)$ for all $n \in N$ . Then let σ small enough, we can get the result. We complete the proof. □

Proof of Theorem 1.1.

Suppose that $m_{n} \in R^{N}$ is a maximum point of $| v_{n} (x) |$ for each $n \in N$ , $\begin{matrix} | v_{n} (m_{n}) | = max_{x \in R^{N}} | v_{n} (x) | . \end{matrix}$ By the definition, $v_{n} (x) = u_{n} (x + y_{n})$ . Then $t_{n} = m_{n} + y_{n}$ is a maximum point of $| u_{n} (x) |$ .

As a direct result of Corollary 5.5, $| v_{n} |_{L^{\infty} (R^{N})} ⩾ η$ for some $η > 0$ , then let η equal to σ in Lemma 5.8, then we have $\begin{matrix} | v_{n} (x) | ⩽ η, \forall | x | ⩾ R, \end{matrix}$ which implies that the sequence ${m_{n}}$ is bounded in $R^{N}$ , together with $ϵ_{n} y_{n} \to z \in A$ by Lemma 5.2, then up to a subsequence, we have $\begin{matrix} ϵ_{n} t_{n} = ϵ_{n} m_{n} + ϵ_{n} y_{n} \to z \in A . \end{matrix}$ Hence $\begin{matrix} lim_{n \to + \infty} A (ϵ_{n} t_{n}) = A (z), z \in A . \end{matrix}$ The proof is completed. □

Footnotes

Acknowledgements

The authors would like to thank the referee for many useful comments which clarify the paper.

C. Ji was partially supported by Natural Science Foundation of Shanghai (20ZR1413900, 18ZR1409100).

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