Existence and Concentration of Sign-Changing Solutions of Quasilinear Choquard Equation

Die Wang; Yuqi Wang; Shaoxiong Chen

doi:10.4236/jamp.2023.114074

Journal of Applied Mathematics and Physics > Vol.11 No.4, April 2023

Existence and Concentration of Sign-Changing Solutions of Quasilinear Choquard Equation

Die Wang, Yuqi Wang, Shaoxiong Chen^*
Department of Mathematics, Yunnan Normal University, Kunming, China.
DOI: 10.4236/jamp.2023.114074 PDF HTML XML 45 Downloads 374 Views

Abstract

In this paper, we study the following quasilinear equation of choquard type:

where A(x,t) is given real functions on R^N × R and with N ≥ 3, 1 < p < N, max{N-2p,1} < α < N, , and ε > 0 is a small parameter, I_α is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique.

Keywords

Quasilinear Choquard Equation, The Method of Invariant Sets of Descending Flow, Truncation, Sign-Changing Solutions

Share and Cite:

Wang, D. , Wang, Y. and Chen, S. (2023) Existence and Concentration of Sign-Changing Solutions of Quasilinear Choquard Equation. Journal of Applied Mathematics and Physics, 11, 1124-1151. doi: 10.4236/jamp.2023.114074.

Keywords:

1. Introduction and Main Results

In this paper, we consider the following quasilinear equation of choquard type

$(\begin{array}{l} - d i v (A (x, u) {| \nabla u |}^{p - 2} \nabla u) + \frac{1}{p} A_{t} (x, u) {| \nabla u |}^{p} + V (ε x) {| u |}^{p - 2} u \\ = (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u, x \in ℝ^{N} \\ u (x) \to 0, | x | \to \infty \end{array}$ (1)

where $A (x, t)$ is given real functions on $ℝ^{N} \times ℝ$ and $A_{t} (x, t) = \frac{\partial}{\partial t} A (x, t)$ with $N \geq 3$ , $1 < p < N$ , $\max {N - 2 p, 1} < α < N$ , $\frac{p (N + α)}{2 N} \leq q < p_{α}^{*} = \frac{p (N + α)}{2 (N - p)}$ , and $ε > 0$ is a small parameter, $I_{α} : ℝ^{N} \to ℝ$ is the Riesz potential defined by

$I_{α} (x) = \frac{Γ (\frac{N - α}{2})}{Γ (\frac{α}{2}) π^{\frac{N}{2}} 2^{α} {| x |}^{N - α}} : = \frac{A_{α}}{{| x |}^{N - α}}$

and $Γ$ is the Gamma function.

In the mathematical literature, very few results are known about Equation (1). Since the classical variational approach fails if $A_{t} (x, t) \equiv 0$ . In reference [1] , the author considered the following quasilinear equation

$- d i v (A (x, u) {| \nabla u |}^{p - 2} \nabla u) + \frac{1}{p} A_{t} (x, u) {| \nabla u |}^{p} + {| u |}^{p - 2} u = g (x, u), x \in ℝ^{N},$ (2)

In fact, the natural functional associated to (2) is

$J (u) = \frac{1}{p} \int_{ℝ^{N}} A (x, u) {| \nabla u |}^{p} d x + \frac{1}{p} \int_{ℝ^{N}} {| u |}^{p} d x - \int_{ℝ^{N}} G (x, u) d x,$

which is not defined in $W^{1, p} (ℝ^{N})$ for a general coefficient $A (x, t)$ in the principal part. Moreover, even if $A (x, t)$ is smooth strictly positive bounded function, the corresponding energy functional is well defined in $W^{1, p} (ℝ^{N})$ , if $A_{t} (x, t) \equiv 0$ it is Gâteaux differentiable only along directions of $W^{1, p} (ℝ^{N}) \cap L^{\infty} (ℝ^{N})$ [1] . More recently, if $Ω$ is a bounded subset of $ℝ^{N}$ a different approach has been developed which exploits the interaction between two different norm on $W_{0}^{1, p} (Ω) \cap L^{\infty} (Ω)$ [2] [3] [4] . So, use the interaction between the norm ${‖ \cdot ‖}_{x}$ and the standard one on $W^{1, p} (ℝ^{N})$ , if $G (x, t)$ has a subcritical growth, we state that $J$ satisfies a weaken version of the Cerami’s variant of the Palais-Smale condition in $X_{r}$ . We note that, in general, $J$ cannot verify the standard Palais-Smale condition, or its Cerami’s variant, as Palais-Smale sequences may converge in the $W^{1, p} (ℝ^{N})$ -norm but be unbounded in $L^{\infty} (ℝ^{N})$ [4] . For this reason, we know that there is a bounded radial solution of Equation (2) under certain assumptions.

For the Choquard equation

$- Δ u + V (x) u = (I_{α} * {| u |}^{p}) {| u |}^{p - 2} u, x \in ℝ^{3} .$ (3)

where $p \in (\frac{2 N - α}{N}, \frac{2 N - α}{N - 2})$ . When the potential V is a positive constant, Lieb [5] obtained the existence and uniqueness of positive radial ground states for (3), Lions [6] established the existence of infinitely many radial solutions, Ma and Zhao [7] studied the radial symmetry and uniqueness of positive ground states for (3) in higher dimension space via the method of moving planes. For more related results, we refer to [2] [8] [9] [10] [11] [12] and references therein.

In this article, we consider the existence of sign-changing solutions for the quasilinear Choquard Equation (1) by using the method of invariant sets of descending flow and the perturbation method. Byeon-Wang type penalization method [13] can be used to deal with multiple localized nodal solutions for semiclassical Schrödinger equations. Additional coercive term [14] can be used to make the perturbed functional has necessary compactness properties in changed space.

From now on, Let $A : ℝ^{N} \times ℝ \to ℝ$ be such that

(A₀) $A (x, t)$ is a $C^{1}$ Carathéodory function, i.e.

$A (\cdot, t) : x \in ℝ^{N} \mapsto A (x, t) \in ℝ$ is measurable for all $t \in ℝ$ ,

$A (x, \cdot) : t \in ℝ \mapsto A (x, t) \in ℝ$ is $C^{1}$ for a.e. $x \in ℝ^{N}$ ;

(A₁) $A (x, t)$ and $A_{t} (x, t)$ are essentially bounded if t is bounded, i.e.

$\sup_{| t | \leq r} | A (\cdot, t) | \in L^{\infty} (ℝ^{N}), \sup_{| t | \leq r} | A_{t} (\cdot, t) | \in L^{\infty} (ℝ^{N}) for any r > 0;$

(A₂) Exists a constant $c_{1}, c_{2} > 0$ such that

$c_{1} \leq A (x, t) \leq c_{2} a .e . x \in ℝ^{N};$

(A₃) Exist constants $R \geq 1$ and $α_{1} > 0$ such that

$p A (x, t) + A_{t} (x, t) t \geq α_{1} A (x, t) a .e . in ℝ^{N}, if | t | \geq R;$

(A₄) Exist constants $μ > p$ and $α_{2} > 0$ such that

$(μ - p) A (x, t) - A_{t} (x, t) t \geq α_{2} A (x, t) a .e . in ℝ^{N}, for all t \in ℝ .$

The potential function V satisfies the following assumptions:

(V₁) $V \in C^{1} (ℝ^{N}, ℝ)$ and there exist constants $b > a > 0$ such that

$a \leq V (x) \leq b, x \in ℝ^{N};$

(V₂) There exists a bounded domain $M \subset ℝ^{N}$ with smooth boundary $\partial M$ such that

$〈 \overset{\leftarrow}{n (x)}, \nabla V (x) 〉 > 0, x \in \partial M .$

where $\overset{\leftarrow}{n (x)}$ is the outer normal of $\partial M$ at x. according to condition (V₂) that

$A = {x \in M | \nabla V (x) = 0} \neq \emptyset .$ (see [15] )

and $A$ is a closed subset of $M$ , without losing generality, we assume that $0 \in A$ .

For any set $B \in ℝ^{N}$ and any $δ > 0$ , we denote

$B^{δ} = {x \in ℝ^{N} | d i s t (x, B) : = \inf_{y \in B} | x - y | < δ},$

$B_{δ} = {x \in ℝ^{N} | δ x \in B} .$

The main results of this paper are as follows:

Theorem 1.1 Assume that $A (x, t)$ satisfy conditions (A₀)-(A₄), the potential function V satisfies (V₁) and (V₂), for any positive integer k, there exists $ε_{k} > 0$ such that if $0 < ε < ε_{k}$ , the problem (1) has at least k pairs of sign-changing solutions $\pm u_{j, ε}, j = 1, 2, \dots, k$ . In addition, for any $δ > 0$ there exist $μ > 0, c = c_{k} > 0$ and $ε_{k} (δ) > 0$ such that if $0 < ε < ε_{k} (δ)$ , then it holds that

$| u_{j, ε} | \leq c \exp {- μ d i s t (x, {(A^{δ})}_{ε})}, j = 1, 2, \dots, k, x \in ℝ^{N} .$

2. Preliminaries

We note that Equation (1) corresponding energy functional is

$I_{ε} (u) = \frac{1}{p} \int_{ℝ^{N}} A (x, u) {| \nabla u |}^{p} d x + \frac{1}{p} \int_{ℝ^{N}} V (ε x) {| u |}^{p} d x - \frac{1}{2 q} \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q} d x$ (4)

We use the penalization method due to Byeon and Wang [13] . Let $ζ \in C^{\infty}$ be a cut-off function, $ζ (t) = 0$ for $t \leq 0$ ; $ζ (t) = 1$ for $t \geq 1$ ; $0 \leq ζ^{'} (t) \leq 2$ and $0 \leq ζ (t) \leq 1$ . Define

$χ_{ε} (x) = (\begin{array}{l} 0, & x \in M_{ε} \\ ε^{- p} ζ (d i s t (x, M_{ε})), & x \notin M_{ε} \end{array}$

In order to overcome the lack of compactness condition, we set the workspace as $X_{ε} = W^{1, p} (ℝ^{N}) \cap L_{ε}^{m} (ℝ^{N})$ , where $L_{ε}^{m} (ℝ^{N})$ is a weighted space defined as

$L_{ε}^{m} (ℝ^{N}) = {u \in L^{m} (ℝ^{N}) | \int_{ℝ^{N}} \exp {(m - p) d i s t (ε x, M)} {| u |}^{m} d x < + \infty},$

The corresponding norm is defined as

${‖ u ‖}_{L_{ε}^{m} (ℝ^{N})} = {(\int_{ℝ^{N}} \exp {(m - p) d i s t (ε x, M)} {| u |}^{m} d x)}^{\frac{1}{m}} .$

which $p < m < \min {2, q}$ if $1 < p < 2$ ; $p < m < q$ if $p \geq 2$ .

Define

${‖ u ‖}_{X_{ε}} = {‖ u ‖}_{W^{1, p} (ℝ^{N})} + {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})} .$

Meanwhile, We add the forced disturbance term such that $I_{ε}$ has the necessary compactness property on $X_{ε}$ , introduce some auxiliary functions. Let $ξ \in C^{\infty} (ℝ, [0,1])$ be a smooth, even function, such that $ξ (t) = 1$ if $| t | \leq 1$ , $ξ (t) = 0$ if $| t | \geq 2$ , and $ξ$ is decreasing in $[1,2]$ . For $ε \in (0,1]$ , $x \in ℝ^{N}$ , $t \in ℝ$ , we define

$b_{ε} (x, t) = ξ (ε \exp {d i s t (ε x, M)} t), m_{ε} (x, t) = \int_{0}^{t} b_{ε} (x, τ) d τ,$

$k_{ε} (x, t) = {(\frac{t}{m_{ε} (x, t)})}^{m - p} {| t |}^{p - 2} t, K_{ε} (x, t) = \int_{0}^{t} k_{ε} (x, τ) d τ .$

Now, we define the perturbation functional:

$\begin{array}{l} Γ_{ε} (u) = \frac{1}{p} \int_{ℝ^{N}} A (x, u) {| \nabla u |}^{p} d x + \frac{1}{p} \int_{ℝ^{N}} E (ε x) {| u |}^{p} d x + σ \int_{ℝ^{N}} K_{ε} (x, u) d x \\ + \frac{1}{p β} {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} - \frac{1}{2 q} \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q} d x, u \in X_{ε} \end{array}$ (5)

where $p < p β < q$ , $E (x) = V (x) - σ$ , $σ > 0$ sufficiently small, E satisfies the assumptions (V₁) and (V₂).

Note that for any $v \in X_{ε}$

$\begin{matrix} 〈 {Γ^{'}}_{ε} (u), v 〉 = \int_{ℝ^{N}} A (x, u) {| \nabla u |}^{p - 2} \nabla u \nabla v d x + \frac{1}{p} \int_{ℝ^{N}} A_{t} (x, u) v {| \nabla u |}^{p} d x \\ + \int_{ℝ^{N}} E (ε x) {| u |}^{p - 2} u v d x + σ \int_{ℝ^{N}} k_{ε} (x, u) v d x \\ + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p - 2} u v d x \\ - \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u v d x \end{matrix}$ (6)

We will use the abstract critical point theorem to prove the existence of the infinitely variable sign solution of Equation (1). However, when verifying the conditions of the theorem, we encounter essential difficulties, that is, we can’t guarantee that the positive cone and the negative cone are flow-invariant, so we follow the idea of literature [14] [16] [17] and use truncation method to truncate the nonlocal term. First, we define the following auxiliary functions:

$b_{λ} (t) = ξ (λ t), m_{λ} (t) = \int_{0}^{t} b_{λ} (τ) d τ,$

$g_{λ} (t) = \frac{m_{λ} (t)}{t}, h_{λ} (t) = g_{λ} (t) + b_{λ} (t) .$

Define the perturbation functional:

$\begin{array}{l} Γ_{ε, λ} (u) = \frac{1}{p} \int_{ℝ^{N}} A (x, u) {| \nabla u |}^{p} d x + \frac{1}{p} \int_{ℝ^{N}} E (ε x) {| u |}^{p} d x + σ \int_{ℝ^{N}} K_{ε} (x, u) d x \\ + \frac{1}{p β} {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} - \frac{1}{2 q} g_{λ} (ψ^{\frac{1}{2}} (u)) ψ (u), u \in X_{ε} \end{array}$ (7)

where $ψ (u) = \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q} d x$ .

By the definition of $b_{λ}, g_{λ}$ , it is easy to prove that $g_{λ} (t)$ satisfies the following properties:

Lemma 2.1 [15] For $t > 0, 0 < λ < 1$ , it holds

(1) $g_{λ} (t) = 1, {g^{'}}_{λ} (t) = 0$ if $0 < t < \frac{1}{λ}$ ;

(2) ${g^{'}}_{λ} (t) t + g_{λ} (t) = b_{λ} (t)$ ;

(3) ${g^{'}}_{λ} (t) t + g_{λ} (t) \leq \frac{c_{λ}}{t}, b_{λ} (t) t \leq g_{λ} (t) t \leq c_{λ}$ , where $c_{λ} = \frac{\int_{0}^{\infty} ξ (τ) d τ}{λ}$ .

$\forall v \in X_{ε}$ , since ${g^{'}}_{λ} (t) t + g_{λ} (t) = b_{λ} (t)$ , we have

$\begin{matrix} 〈 {Γ^{'}}_{ε, λ} (u), v 〉 = \int_{ℝ^{N}} A (x, u) {| \nabla u |}^{p - 2} \nabla u \nabla v d x + \frac{1}{p} \int_{ℝ^{N}} A_{t} (x, u) v {| \nabla u |}^{p} d x \\ + \int_{ℝ^{N}} E (ε x) {| u |}^{p - 2} u v d x + σ \int_{ℝ^{N}} k_{ε} (x, u) v d x \\ + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p - 2} u v d x \\ - \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u)) \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u v d x . \end{matrix}$ (8)

According to Hardy-Littlewood-Sobolev inequality and Sobolev inequality

$ψ^{\frac{1}{2}} (u) \leq c {‖ u ‖}_{W^{1, p} (ℝ^{N})}^{q} .$

Therefore, when ${‖ u ‖}_{W^{1, p} (ℝ^{N})} \leq {(\frac{1}{c λ})}^{\frac{1}{q}}$ , there is $Γ_{ε, λ} (u) = Γ_{ε} (u)$ and ${Γ^{'}}_{ε, λ} (u) = {Γ^{'}}_{ε} (u)$ ; when $| u (x) | \leq ε^{- 1} \exp {- d i s t (ε x, M)}$ and ${(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+} = 0$ , there is $Γ_{ε} (u) = I_{ε} (u)$ and ${Γ^{'}}_{ε} (u) = {I^{'}}_{ε} (u)$ . Hence, finding the solution of the Equation (4) translates into finding the critical point of $Γ_{ε, λ} (u)$ .

We describe the abstract critical point theorem in detail below [15] .

Let X be a Banach space and f be an even $C^{1}$ functional on X. Let $P, Q$ are two open convex sets of X, $Q = - P$ . Set

$W = P \cup Q, Σ = \partial P \cap \partial Q .$

Assume

(I₁) f satisfies the (PS) condition.

(I₂) $c^{*} = \inf_{x \in Σ} f (x) > 0$ . And assume there exists an odd continuous map $F : X \to X$ satisfying the following:

(F₁) given $c_{0}, b_{0} > 0$ , there exists $b = b (c_{0}, b_{0}) > 0$ such that if $‖ f^{'} (x) ‖ \geq b_{0}$ , $| f (x) | \leq c_{0}$ , then

$〈 f^{'} (x), x - F x 〉 \geq b {‖ x - F x ‖}_{X} > 0.$

(F₂) $F (\partial P) \subset P$ , $F (\partial Q) \subset Q$ .

Define

$Γ_{j} = {E | E \subset X : E compact, - E = E, γ (E \cap η^{- 1} (Σ)) \geq j for η \in Λ},$

$Λ = {η | η \in C (X, X) : η odd, η (P) \subset P, η (Q) \subset Q, η (x) = x if f (x) < 0} .$

where $γ$ is the genus of symmetric sets, defined by

$γ (E) = \inf {n | there exists an odd map η : E \to ℝ^{n} \ {0}} .$

Assume

(Γ) $Γ_{j}, j = 1, 2, \dots$ is nonempty.

We define

$c_{j} = \inf_{E \in Γ_{j}} \sup_{x \in E \ W} f (x), j = 1, 2, \dots,$

$K_{c} = {x | f^{'} (x) = 0, f (x) = c}, K_{c}^{*} = K_{c} \ W .$

Theorem 2.2 ( [18] , Theorem 2.5) Assume (I₁), (I₂), (F₁), (F₂), (Γ) hold, then

(1) $c_{j} \geq c^{*}$ , $K_{c_{j}}^{*} \neq \emptyset$ ,

(2) $c_{j} \to \infty$ , for $j \to \infty$ .

(3) $γ (K_{c}^{*}) \geq k$ , if $c_{j} = c_{j + 1} = \dots = c_{j + k - 1} = c$ .

3. Existence of Sign-Changing Critical Points

In this section, we first introduce some important properties of auxiliary functions, then prove that $Γ_{ε, λ}$ satisfies the (PS) condition, and then use Theorem B to prove the existence of the critical point of $Γ_{ε, λ}$ .

Lemma 3.1 ( [19] , Lemma 2.2), ( [20] , Lemma 2.1) For $x \in ℝ^{N}$

(1) $0 \leq b_{ε} (x, t) \leq \frac{m_{ε} (x, t)}{t} \leq 1$ ;

(2) If $| t | < ε^{- 1} \exp {- d i s t (ε x, M)}$ , then $m_{ε} (x, t) = t$ ;

If $ε^{- 1} \exp {- d i s t (ε x, M)} \leq | t | \leq 2 ε^{- 1} \exp {- d i s t (ε x, M)}$ , then $ε^{- 1} \exp {- d i s t (ε x, M)} \leq | m_{ε} (x, t) | \leq c ε^{- 1} \exp {- d i s t (ε x, M)}$ ;

If $| t | > 2 ε^{- 1} \exp {- d i s t (ε x, M)}$ , then $| m_{ε} (x, t) | = c ε^{- 1} \exp {- d i s t (ε x, M)}$ ; where $c = \int_{0}^{\infty} ξ (τ) d τ$ ;

(3) $\begin{array}{l} c_{1} (1 + ε^{m - p} \exp {(m - p) d i s t (ε x, M)} {| t |}^{m - p}) {| t |}^{p - 2} t \leq k_{ε} (x, t) \\ \leq c_{2} (1 + ε^{m - p} \exp {(m - p) d i s t (ε x, M)} {| t |}^{m - p}) {| t |}^{p - 2} t \end{array}$ ;

(4) $\frac{1}{m} t k_{ε} (x, t) \leq K_{ε} (x, t) \leq \frac{1}{p} t k_{ε} (x, t)$ ;

(5) $(k_{ε} (x, t_{1}) - k_{ε} (x, t_{2})) (t_{1} - t_{2}) \geq c ε^{m - p} \exp {(m - p) d i s t (ε x, M)} {| t_{1} - t_{2} |}^{m}$ , $p \geq 2$ ;

$(k_{ε} (x, t_{1}) - k_{ε} (x, t_{2})) (t_{1} - t_{2}) \geq c ε^{m - p} \exp {(m - p) d i s t (ε x, M)} \frac{{| t_{1} - t_{2} |}^{2}}{{| t_{1} |}^{2 - m} + {| t_{2} |}^{2 - m}}$ , $1 < p < 2$ ;

(6) $\begin{array}{l} | k_{ε} (x, t_{1}) - k_{ε} (x, t_{2}) | \\ \leq c ({| t_{1} |}^{p - 2} + {| t_{2} |}^{p - 2} + ε^{m - p} \exp {(m - p) d i s t (ε x, M)} ({| t_{1} |}^{m - 2} + {| t_{2} |}^{m - 2})) | t_{1} - t_{2} | \end{array}$ , $p \geq 2$ ;

$\begin{array}{l} | k_{ε} (x, t_{1}) - k_{ε} (x, t_{2}) | \\ \leq c ({| t_{1} - t_{2} |}^{p - 1} + ε^{m - p} \exp {(m - p) d i s t (ε x, M)} {| t_{1} - t_{2} |}^{m - 1}) \end{array}$ , $1 < p < 2$ .

Lemma 3.2 Let ${u_{n}} \subset X_{ε}$ be a Palais-Smale sequence of the functional $Γ_{ε, λ}$ , then ${u_{n}}$ is bounded in $X_{ε}$ .

Proof: According to (7), (8) and assumptions (A₄), (A₂) and Lemma 3.1 (4) (3), we have

$\begin{array}{l} μ Γ_{ε, λ} (u_{n}) - 〈 {Γ^{'}}_{ε, λ} (u_{n}), u_{n} 〉 \\ = μ [\frac{1}{p} \int_{ℝ^{N}} A (x, u_{n}) {| \nabla u_{n} |}^{p} d x + \frac{1}{p} \int_{ℝ^{N}} E (ε x) {| u_{n} |}^{p} d x + σ \int_{ℝ^{N}} K_{ε} (x, u_{n}) d x \\ + \frac{1}{p β} {(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β} - \frac{1}{2 q} g_{λ} (ψ^{\frac{1}{2}} (u_{n})) ψ (u_{n})] \\ - \int_{ℝ^{N}} A (x, u_{n}) {| \nabla u_{n} |}^{p} d x - \frac{1}{p} \int_{ℝ^{N}} A_{t} (x, u_{n}) u_{n} {| \nabla u_{n} |}^{p} d x - \int_{ℝ^{N}} E (ε x) {| u_{n} |}^{p} d x \\ - {(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x \end{array}$

$\begin{array}{l} - σ \int_{ℝ^{N}} k_{ε} (x, u_{n}) u_{n} d x + \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u_{n})) ψ (u_{n}) \\ = \frac{1}{p} \int_{ℝ^{N}} ((μ - p) A (x, u_{n}) - A_{t} (x, u_{n}) u_{n}) {| \nabla u_{n} |}^{p} d x + \frac{μ - p}{p} \int_{ℝ^{N}} E (ε x) {| u_{n} |}^{p} d x \\ + σ \int_{ℝ^{N}} (μ K_{ε} (x, u_{n}) - k_{ε} (x, u_{n}) u_{n}) d x + \frac{μ}{p β} {(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β} \\ - {(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x \\ + \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u_{n})) ψ (u_{n}) - \frac{μ}{2 q} g_{λ} (ψ^{\frac{1}{2}} (u_{n})) ψ ( u n ) \end{array}$

$\begin{array}{l} \geq \frac{α_{2}}{p} \int_{ℝ^{N}} A (x, u_{n}) {| \nabla u_{n} |}^{p} d x + \frac{μ - p}{p} \int_{ℝ^{N}} E (ε x) {| u_{n} |}^{p} d x \\ + σ (\frac{μ}{m} - 1) \int_{ℝ^{N}} k_{ε} (x, u_{n}) u_{n} d x + c {(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β} - c \\ \geq \frac{c_{1} α_{2}}{p} \int_{ℝ^{N}} {| \nabla u_{n} |}^{p} d x + \frac{μ - p}{p} \int_{ℝ^{N}} E (ε x) {| u_{n} |}^{p} d x + c {(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β} \\ - c + c \int_{ℝ^{N}} c_{1} (1 + ε^{m - p} \exp {(m - p) d i s t (ε x, M)} {| u_{n} |}^{m - p}) {| u_{n} |}^{p} d x \end{array}$

$\begin{array}{l} \geq c ({‖ u_{n} ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u_{n} ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}) + c {(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β} - c \\ \geq c {‖ u_{n} ‖}_{X_{ε} (ℝ^{N})}^{p} + c {(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β} - c \end{array}$

Thus, The (PS) sequence ${u_{n}}$ of functional $Γ_{ε, λ}$ is bounded in $X_{ε}$ . $■$

Lemma 3.3 The embedding $X_{ε}$ ↪ $L^{r} (ℝ^{N}) (1 \leq r < p^{*})$ is compact.

Proof: Let ${u_{n}}$ be the (PS) sequence of functional $Γ_{ε, λ}$ , $u_{n} \subset X_{ε}$ satisfy $Γ_{ε, λ} (u_{n}) \to c$ , ${Γ^{'}}_{ε, λ} (u_{n}) \to 0$ in ${(X_{ε})}^{'}$ , by Lemma 3.2, there is a constant ${\hat{η}}_{L} > 0$ independent of $ε$ such that ${‖ u_{n} ‖}_{ε} \leq {\hat{η}}_{L}$ and ${(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β} \leq {\hat{η}}_{L}$ . Up to subsequences, suppose $u_{n} ⇀ u$ in $X_{ε}$ and $u_{n} \to u$ in $L_{l o c}^{r} (ℝ^{N}) (1 \leq r < p^{*})$ . First prove $u_{n} \to u$ in $L^{1} (ℝ^{N})$ , for any $R > 0$ that

$\begin{array}{l} \int_{ℝ^{N} \ B (0, R)} | u | d x \\ = \int_{ℝ^{N} \ B (0, R)} \exp {\frac{m - p}{m} d i s t (ε x, M)} \cdot \exp {- \frac{m - p}{m} d i s t (ε x, M)} \cdot | u | d x \\ \leq {(\int_{ℝ^{N} \ B (0, R)} \exp {(m - p) d i s t (ε x, M)} {| u |}^{m} d x)}^{\frac{1}{m}} \\ \cdot {(\int_{ℝ^{N} \ B (0, R)} \exp {- \frac{m - p}{m - 1} d i s t (ε x, M)} d x)}^{\frac{m - 1}{m}} \\ \leq {‖ u ‖}_{L_{ε}^{M} (ℝ^{N})} {(\int_{ℝ^{N} \ B (0, R)} \exp {- \frac{m - p}{m - 1} d i s t (ε x, M)} d x)}^{\frac{m - 1}{m}} \\ = o_{R} (1), \end{array}$

Hence

$\begin{matrix} \int_{ℝ^{N}} | u_{n} - u | d x = \int_{B (0, R)} | u_{n} - u | d x + \int_{ℝ^{N} \ B (0, R)} | u_{n} - u | d x \\ = o_{n} (1) + o_{R} (1) \to 0. \end{matrix}$

For $1 < r < p^{*}$ ,

$\begin{matrix} \int_{ℝ^{N}} {| u_{n} - u |}^{r} d x = \int_{ℝ^{N}} {| u_{n} - u |}^{r θ + (1 - θ) r} d x \\ \leq {(\int_{ℝ^{N}} {| u_{n} - u |}^{r θ \cdot \frac{1}{r θ}} d x)}^{r θ} \cdot {(\int_{ℝ^{N}} {| u_{n} - u |}^{(1 - θ) r \cdot \frac{p^{*}}{(1 - θ) r}} d x)}^{\frac{(1 - θ) r}{p^{*}}} \\ \leq c {(\int_{ℝ^{N}} | u_{n} - u | d x)}^{r θ} \to 0. \end{matrix}$

where $0 < θ < 1$ , $\frac{1}{r} = θ + \frac{1 - θ}{p^{*}}$ . $■$

Lemma 3.4 For every $ε > 0$ , $Γ_{ε, λ}$ satisfies the Palais-Smale condition.

Proof: Let ${u_{n}} \subset X_{ε}$ be a Palais-Smale sequence of the functional $Γ_{ε, λ}$ , so $| Γ_{ε, λ} (u_{n}) | \leq c$ and ${Γ^{'}}_{ε, λ} (u_{n}) \to 0 (n \to \infty)$ , We will prove that ${u_{n}}$ has a convergent subsequence in $X_{ε}$ . According to Lemma 3.2, $u_{n} ⇀ u$ in $X_{ε}$ , then by assuming A₂, A₁, Hardy-Littlewood-Sobolev inequality, Hölder inequality and Lemma 3.3, there are

$\begin{matrix} o (1) = 〈 {Γ^{'}}_{ε, λ} (u_{n}) - {Γ^{'}}_{ε, λ} (u), u_{n} - u 〉 \\ = \int_{ℝ^{N}} (A (x, u_{n}) {| \nabla u_{n} |}^{p - 2} \nabla u_{n} - A (x, u) {| \nabla u |}^{p - 2} \nabla u, \nabla u_{n} - \nabla u) d x \\ + \frac{1}{p} \int_{ℝ^{N}} (A_{t} (x, u_{n}) {| \nabla u_{n} |}^{p} - A_{t} (x, u) {| \nabla u |}^{p}, u_{n} - u) d x \\ + \int_{ℝ^{N}} E (ε x) ({| u_{n} |}^{p - 2} u_{n} - {| u |}^{p - 2} u, u_{n} - u) d x \\ + σ \int_{ℝ^{N}} (k_{ε} (x, u_{n}) - k_{ε} (x, u)) (u_{n} - u) d x \end{matrix}$

$\begin{array}{l} + {(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p - 2} u_{n} (u_{n} - u) d x \\ - {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p - 2} u (u_{n} - u) d x \\ - \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u_{n})) \int_{ℝ^{N}} (I_{α} * {| u_{n} |}^{q}) ({| u_{n} |}^{q - 2} u_{n} (u_{n} - u)) d x \\ + \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u)) \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) ({| u |}^{q - 2} u (u_{n} - u)) d x \end{array}$

$\begin{array}{l} \geq c_{1} \int_{ℝ^{N}} ({| \nabla u_{n} |}^{p - 2} \nabla u_{n} - {| \nabla u |}^{p - 2} \nabla u, \nabla u_{n} - \nabla u) d x \\ + \int_{ℝ^{N}} E (ε x) ({| u_{n} |}^{p - 2} u_{n} - {| u |}^{p - 2} u, u_{n} - u) d x \\ + σ \int_{ℝ^{N}} (k_{ε} (x, u_{n}) - k_{ε} (x, u)) (u_{n} - u) d x + o (1) . \end{array}$

Thus

$\int_{ℝ^{N}} ({| \nabla u_{n} |}^{p - 2} \nabla u_{n} - {| \nabla u |}^{p - 2} \nabla u, \nabla u_{n} - \nabla u) d x \to 0;$

$\int_{ℝ^{N}} ({| u_{n} |}^{p - 2} u_{n} - {| u |}^{p - 2} u, u_{n} - u) d x \to 0;$

$\int_{ℝ^{N}} (k_{ε} (x, u_{n}) - k_{ε} (x, u)) (u_{n} - u) d x \to 0.$

In the following proof process, the following basic inequalities will be used [21] .

$\forall y, z \in ℝ^{N}$ , when $p \geq 2$ there is

$(\begin{array}{l} ({| y |}^{p - 2} y - {| z |}^{p - 2} z, y - z) \geq c {| y - z |}^{p}, \\ | {| y |}^{p - 2} y - {| z |}^{p - 2} z | \leq c ({| y |}^{p - 2} + {| z |}^{p - 2}) | y - z |; \end{array}$ (9)

when $1 < p < 2$ there is

$(\begin{array}{l} ({| y |}^{p - 2} y - {| z |}^{p - 2} z, y - z) \geq c {| y - z |}^{2} {({| y |}^{2 - p} + {| z |}^{2 - p})}^{- 1}, \\ | {| y |}^{p - 2} y - {| z |}^{p - 2} z | \leq c {| y - z |}^{p - 1} . \end{array}$ (10)

For $p \geq 2$ , by (9) we have

$\int_{ℝ^{N}} {| \nabla (u_{n} - u) |}^{p} \leq c \int_{ℝ^{N}} ({| \nabla u_{n} |}^{p - 2} \nabla u_{n} - {| \nabla u |}^{p - 2} \nabla u, \nabla u_{n} - \nabla u) d x \to 0,$

$\int_{ℝ^{N}} {| u_{n} - u |}^{p} \leq c \int_{ℝ^{N}} ({| u_{n} |}^{p - 2} u_{n} - {| u |}^{p - 2} u, u_{n} - u) d x \to 0.$

In addition, it follows from Lemma 3.1 (5) that

$\begin{array}{l} \int_{ℝ^{N}} \exp {(m - p) d i s t (ε x, M)} {| u_{n} - u |}^{m} \\ \leq c \int_{ℝ^{N}} (k_{ε} (x, u_{n}) - k_{ε} (x, u)) (u_{n} - u) d x \to 0. \end{array}$

For $1 < p < 2$ , by (10) we have

$\begin{array}{l} \int_{ℝ^{N}} {| \nabla (u_{n} - u) |}^{p} \\ \leq c \int_{ℝ^{N}} {({| \nabla u_{n} |}^{p - 2} \nabla u_{n} - {| \nabla u |}^{p - 2} \nabla u, \nabla u_{n} - \nabla u)}^{\frac{p}{2}} {({| \nabla u_{n} |}^{2 - p} + {| \nabla u |}^{2 - p})}^{\frac{p}{2}} d x \\ \leq c {(\int_{ℝ^{N}} ({| \nabla u_{n} |}^{p - 2} \nabla u_{n} - {| \nabla u |}^{p - 2} \nabla u, \nabla u_{n} - \nabla u) d x)}^{\frac{p}{2}} \\ \cdot {(\int_{ℝ^{N}} {({| \nabla u_{n} |}^{2 - p} + {| \nabla u |}^{2 - p})}^{\frac{p}{2 - p}} d x)}^{\frac{2 - p}{2}} \\ \leq c {(\int_{ℝ^{N}} ({| \nabla u_{n} |}^{p - 2} \nabla u_{n} - {| \nabla u |}^{p - 2} \nabla u, \nabla u_{n} - \nabla u) d x)}^{\frac{p}{2}} \to 0, \end{array}$

Similarly,

$\int_{ℝ^{N}} {| u_{n} - u |}^{p} \leq c {(\int_{ℝ^{N}} ({| u_{n} |}^{p - 2} u_{n} - {| u |}^{p - 2} u, u_{n} - u) d x)}^{\frac{p}{2}} \to 0.$

It follows from Lemma 3.1 (5) that

$\begin{array}{l} (k_{ε} (x, u_{n}) - k_{ε} (x, u)) (u_{n} - u) \\ \geq c ε^{m - p} \exp {(m - p) d i s t (ε x, M)} \frac{{| u_{n} - u |}^{2}}{{| u_{n} |}^{2 - m} + {| u |}^{2 - m}} . \end{array}$

then

$\begin{array}{l} \exp {(m - p) d i s t (ε x, M)} {| u_{n} - u |}^{m} \\ \leq \exp {\frac{(2 - m) (m - p)}{2} d i s t (ε x, M)} \cdot {({| u_{n} |}^{2 - m} + {| u |}^{2 - m})}^{\frac{m}{2}} \\ \cdot {[(k_{ε} (x, u_{n}) - k_{ε} (x, u)) (u_{n} - u)]}^{\frac{m}{2}} . \end{array}$

Thus

$\begin{array}{l} \int_{ℝ^{N}} \exp {(m - p) d i s t (ε x, M)} {| u_{n} - u |}^{m} \\ \leq c {(\int_{ℝ^{N}} \exp {(m - p) d i s t (ε x, M)} {({| u_{n} |}^{2 - m} + {| u |}^{2 - m})}^{\frac{m}{2 - m}} d x)}^{\frac{2 - m}{2}} \\ \cdot {(\int_{ℝ^{N}} (k_{ε} (x, u_{n}) - k_{ε} (x, u)) (u_{n} - u) d x)}^{\frac{m}{2}} \\ \leq c {(\int_{ℝ^{N}} (k_{ε} (x, u_{n}) - k_{ε} (x, u)) (u_{n} - u) d x)}^{\frac{m}{2}} \to 0. \end{array}$

So, $u_{n} \to u$ in $X_{ε}$ , $Γ_{ε, λ}$ satisfies the Palais-Smale condition. $■$

Next, we will prove the existence of the critical point of the functional $Γ_{ε, λ}$ by using the descending invariant set method Theorem 2.2. Before verifying the condition of Theorem 2.2, we will give a few important Lemmas:

Lemma 3.5 Let $\max {N - 2 p, 1} < α < N$ , $p < q < p_{α}^{*}$ , if $u_{n} \subset W^{1, p} (ℝ^{N})$ , $u_{n} ⇀ u$ in $W^{1, p} (ℝ^{N})$ , and $u_{n} (x) \to u (x)$ a.e. $x \in ℝ^{N}$ , then there is a subcolumn still marked as ${u_{n}}$ , which satisfies

(1) $\int_{ℝ^{N}} (I_{α} * {| u_{n} |}^{q}) {| u_{n} |}^{q} = \int_{ℝ^{N}} (I_{α} * {| u_{n} - u |}^{q}) {| u_{n} - u |}^{q} + \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q} + o_{n} (1),$

(11)

(2) $\begin{array}{l} \int_{ℝ^{N}} [(I_{α} * {| u_{n} |}^{q}) {| u_{n} |}^{q - 2} u_{n} - (I_{α} * {| u_{n} - u |}^{q}) {| u_{n} - u |}^{q - 2} (u_{n} - u) - (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u] φ \\ = o_{n} (1) {‖ φ ‖}_{W^{1, p} (ℝ^{N})} . \end{array}$

(12)

where $φ \in c_{o}^{\infty} (ℝ^{N})$ , as $n \to \infty$ , $o_{n} (1) \to 0$ .

To prove Lemma 3.5, we also need the following results:

Lemma 3.6 ( [22] , Theorem 4.2.7) Let $Ω \subseteq ℝ^{N}$ be a domain and ${u_{n}}$ is bounded in $L^{q} (Ω)$ for some $q > 1$ , If $u_{n} (x) \to u (x)$ a.e. $x \in Ω$ , then $u_{n} ⇀ u$ in $L^{q} (Ω)$ .

Lemma 3.7 ( [23] , Theorem 2.5) If ${u_{n}}$ is bounded in $W^{1, p} (ℝ^{N})$ , $u_{n} ⇀ u$ in $W^{1, p} (ℝ^{N})$ and $u_{n} (x) \to u (x)$ a.e. $x \in ℝ^{N}$ , then for $p < q < p_{α}^{*}$

(1) $\int_{ℝ^{N}} {| {| u_{n} |}^{q} - {| u_{n} - u |}^{q} - {| u |}^{q} |}^{\frac{2 N}{N + α}} d x \to 0,$

(2) $\int_{ℝ^{N}} {| {| u_{n} |}^{q} + {| u_{n} - u |}^{q} - {| u |}^{q} |}^{\frac{2 N}{N + α}} d x \to 0,$

(3) $\int_{ℝ^{N}} {| {| u_{n} |}^{q - 2} u_{n} - {| u_{n} - u |}^{q - 2} (u_{n} - u) - {| u |}^{q - 2} u |}^{\frac{2 N p}{(N + α) p - 2 N + 2 p}} d x \to 0.$

Lemma 3.8 ( [24] , Theorem 2.6) Let $0 < α < N$ , $s \in (1, \frac{N}{α})$ , and let

${u_{n}} \subset L^{1} (ℝ^{N}) \cap L^{s} (ℝ^{N})$ be bounded and such that, up to a subsequence, for any bounded domain $Ω \subset ℝ^{N}$ , $u_{n} \to 0$ in $L^{s} (Ω)$ as $n \to \infty$ . Then, up to a subsequence if necessary, $(I_{α} * u_{n}) (x) \to 0$ a.e. $x \in ℝ^{N}$ as $n \to \infty$ .

Proof of Lemma 3.5:

(1) We note that:

$\begin{array}{l} \int_{ℝ^{N}} (I_{α} * {| u_{n} |}^{q}) {| u_{n} |}^{q} - (I_{α} * {| u_{n} - u |}^{q}) {| u_{n} - u |}^{q} - (I_{α} * {| u |}^{q}) {| u |}^{q} \\ = \int_{ℝ^{N}} (I_{α} * ({| u_{n} |}^{q} + {| u_{n} - u |}^{q})) ({| u_{n} |}^{q} - {| u_{n} - u |}^{q}) - (I_{α} * {| u |}^{q}) {| u |}^{q}, \end{array}$

By the Hardy-Littlewood-Sobolev inequality, for sufficiently small $δ > 0$ , there exists $K_{1} > 0$ , which makes

$| \int_{Ω_{1}} (I_{α} * {| u |}^{q}) {| u |}^{q} | \leq \frac{δ}{6}; Ω_{1} : = {x \in ℝ^{N} : | u (x) | \geq K_{1}} .$

Using the Hardy-Littlewood-Sobolev inequality again, we have

$\begin{array}{l} | \int_{Ω_{1}} (I_{α} * ({| u_{n} |}^{q} + {| u_{n} - u |}^{q})) ({| u_{n} |}^{q} - {| u_{n} - u |}^{q}) | \\ \leq C (N, α) {(\int_{ℝ^{N}} {| {| u_{n} |}^{q} + {| u_{n} - u |}^{q} |}^{\frac{2 N}{N + α}})}^{\frac{N + α}{2 N}} \cdot {(\int_{Ω_{1}} {| {| u_{n} |}^{q} - {| u_{n} - u |}^{q} |}^{\frac{2 N}{N + α}})}^{\frac{N + α}{2 N}} \\ \leq C (N, α) {(\int_{Ω_{1}} {| {| u_{n} |}^{q} - {| u_{n} - u |}^{q} |}^{\frac{2 N}{N + α}})}^{\frac{N + α}{2 N}} \\ \leq C (N, α) {(\int_{Ω_{1}} {| u |}^{\frac{2 N}{N + α}})}^{\frac{N + α}{2 N}}, \end{array}$

For the above $δ$ , when $K_{1}$ is large enough, there is

$| \int_{Ω_{1}} (I_{α} * ({| u_{n} |}^{q} + {| u_{n} - u |}^{q})) ({| u_{n} |}^{q} - {| u_{n} - u |}^{q}) | \leq \frac{δ}{6} .$

Similary, let $Ω_{2} : = {x \in ℝ^{N} : | x | \geq R} \ Ω_{1}$ , let $R > 0$ be large enough so that

$| \int_{Ω_{2}} (I_{α} * {| u |}^{q}) {| u |}^{q} | \leq \frac{δ}{6},$

and

$| \int_{Ω_{2}} (I_{α} * ({| u_{n} |}^{q} + {| u_{n} - u |}^{q})) ({| u_{n} |}^{q} - {| u_{n} - u |}^{q}) | \leq \frac{δ}{6} .$

For $K_{2} > K_{1}$ , let $Ω_{3} (n) : = {x \in ℝ^{N} : | u (x) | \geq K_{2}} \ (Ω_{1} \cup Ω_{2})$ . If $Ω_{3} (n) \neq ϕ$ , then $\forall x \in Ω_{3} (n)$ , there is $| u (x) | < K_{1}$ and $| x | < R$ . Observed, $u_{n} (x) \to u (x)$ a.e. $x \in Ω$ . By Severini-Egoroff theorem, $u_{n}$ converges to u in $B_{R} (0)$ , thus $| Ω_{3} (n) | \to 0$ . So, for a large enough n

$| \int_{Ω_{3} (n)} (I_{α} * {| u |}^{q}) {| u |}^{q} | \leq \frac{δ}{6},$

and

$| \int_{Ω_{3} (n)} (I_{α} * ({| u_{n} |}^{q} + {| u_{n} - u |}^{q})) ({| u_{n} |}^{q} - {| u_{n} - u |}^{q}) | \leq \frac{δ}{6} .$

Finally, we estimate that

$\int_{Ω_{4} (n)} (I_{α} * ({| u_{n} |}^{q} + {| u_{n} - u |}^{q})) ({| u_{n} |}^{q} - {| u_{n} - u |}^{q}) - (I_{α} * {| u |}^{q}) {| u |}^{q}$

where $Ω_{4} (n) = ℝ^{N} \ (Ω_{1} \cup Ω_{2} \cup Ω_{3} (n))$ , $Ω_{4} (n) \subset B_{R} (0)$ .

It follows from Lebesgue dominated convergence theorem that

$\lim_{n \to \infty} \int_{Ω_{4} (n)} {| u_{n} - u |}^{\frac{2 N q}{N + α}} = 0 and \lim_{n \to \infty} \int_{Ω_{4} (n)} {| {| u_{n} |}^{q} - {| u |}^{q} |}^{\frac{2 N}{N + α}} = 0.$

This means that by the Hardy-Littlewood-Sobolev inequality,

$\begin{array}{l} | \int_{Ω_{4} (n)} (I_{α} * ({| u_{n} |}^{q} + {| u_{n} - u |}^{q})) {| u_{n} - u |}^{q} | \\ \leq C (N, α) {(\int_{Ω_{4} (n)} {| u_{n} - u |}^{\frac{2 N q}{N + α}})}^{\frac{N + α}{2 N}} \to 0, \end{array}$

$\begin{array}{l} | \int_{Ω_{4} (n)} (I_{α} * ({| u_{n} |}^{q} + {| u_{n} - u |}^{q})) ({| u_{n} |}^{q} - {| u |}^{q}) | \\ \leq C (N, α) {(\int_{Ω_{4} (n)} {| {| u_{n} |}^{q} - {| u |}^{q} |}^{\frac{2 N}{N + α}})}^{\frac{N + α}{2 N}} \to 0. \end{array}$

Let $H_{n} = {| u_{n} |}^{q} + {| u_{n} - u |}^{q} - {| u |}^{q}$ , then

$\begin{array}{l} \lim_{n \to \infty} \int_{Ω_{4} (n)} (I_{α} * ({| u_{n} |}^{q} + {| u_{n} - u |}^{q})) ({| u_{n} |}^{q} - {| u_{n} - u |}^{q}) - (I_{α} * {| u |}^{q}) {| u |}^{q} \\ = \lim_{n \to \infty} \int_{Ω_{4} (n)} (I_{α} * H_{n}) {| u |}^{q} . \end{array}$

Because $H_{n}$ is bounded in $L^{\frac{2 N}{N + α}} (ℝ^{N})$ and $H_{n} \to 0$ a.e. $x \in ℝ^{N}$ , by Lemma 3.6, $H_{n} ⇀ 0$ in $L^{\frac{2 N}{N + α}} (ℝ^{N})$ , thus $I_{α} * H_{n} ⇀ 0$ in $L^{\frac{2 N}{N - α}} (ℝ^{N})$ , so

$\lim_{n \to \infty} \int_{Ω_{4} (n)} (I_{α} * H_{n}) {| u |}^{q} \to 0.$

Thus, summing up, we obtain that

$\underset{n \to \infty}{\lim \sup} | \int_{ℝ^{N}} (I_{α} * {| u_{n} |}^{q}) {| u_{n} |}^{q} - (I_{α} * {| u_{n} - u |}^{q}) {| u_{n} - u |}^{q} - (I_{α} * {| u |}^{q}) {| u |}^{q} | \leq δ .$

By the arbitrariness of $δ$ , conclusion (1) is established.

(2) According to Lemma 3.7, we have

$\lim_{n \to \infty} \int_{ℝ^{N}} {| {| u_{n} |}^{q} - {| u_{n} - u |}^{q} - {| u |}^{q} |}^{\frac{2 N}{N + α}} d y = 0,$

thus

$| \int_{ℝ^{N}} (I_{α} * ({| u_{n} |}^{q} - {| u_{n} - u |}^{q} - {| u |}^{q})) V_{n} φ | = o_{n} (1) {‖ φ ‖}_{W^{1, p} (ℝ^{N})},$ (13)

where $V_{n} = {| u_{n} |}^{q - 2} u_{n}, {| u_{n} - u |}^{q - 2} (u_{n} - u)$ or ${| u |}^{q - 2} u$ .

Similarly,

$\lim_{n \to \infty} \int_{ℝ^{N}} {| {| u_{n} |}^{q - 2} u_{n} - {| u_{n} - u |}^{q - 2} (u_{n} - u) - {| u |}^{q - 2} u |}^{\frac{2 N p}{(N + α) p - 2 N + 2 p}} d x = 0,$

thus

$| \int_{ℝ^{N}} (I_{α} * W_{n}) ({| u_{n} |}^{q - 2} u_{n} - {| u_{n} - u |}^{q - 2} (u_{n} - u) - {| u |}^{q - 2} u) φ | = o_{n} (1) {‖ φ ‖}_{W^{1, p} (ℝ^{N})},$ (14)

where $W_{n} = {| u_{n} |}^{q}, {| u_{n} - u |}^{q}$ or ${| u |}^{q}$ .

The direct calculation of (13) + (14) is

$\begin{array}{l} \int_{ℝ^{N}} [(I_{α} * {| u_{n} |}^{q}) {| u_{n} |}^{q - 2} u_{n} - (I_{α} * {| u_{n} - u |}^{q}) {| u_{n} - u |}^{q - 2} (u_{n} - u) - (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u] φ \\ = \int_{ℝ^{N}} (I_{α} * {| u_{n} - u |}^{q}) {| u |}^{q - 2} u φ + \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u_{n} - u |}^{q - 2} (u_{n} - u) φ \\ + o_{n} (1) {‖ φ ‖}_{W^{1, p} (ℝ^{N})} . \end{array}$

According to Rellich theorem, there is a subcolumn, which may as well still be recorded as ${u_{n}}$ , for any bounded region $Ω \subset ℝ^{N}$ , we have ${| u_{n} - u |}^{q} \to 0$ in $L^{\frac{2 N}{N + α}} (Ω)$ . By Lemma 3.8, $I_{α} * {| u_{n} - u |}^{q} \to 0$ a.e. $x \in ℝ^{N}$ , from Hardy-Littlewood-Sobolev inequality it follows that

$\sup_{n} {‖ I_{α} * {| u_{n} - u |}^{q} ‖}_{L^{\frac{2 N}{N - α}} (ℝ^{N})} \leq \sup_{n} {‖ | u_{n} - u | ‖}_{L^{\frac{2 N q}{N + α}} (ℝ^{N})} < \infty .$

By Lemma 3.6, we have ${‖ I_{α} * {| u_{n} - u |}^{q} ‖}^{\frac{N p}{N p - N + p}} ⇀ 0$ in $L^{\frac{2 N p - 2 N + 2 p}{(N - α) p}} (ℝ^{N})$ , with $\frac{2 N p - 2 N + 2 p}{(N - α) p} > 1$ .

Because ${| u |}^{\frac{N p (q - 1)}{N p - N + p}} \in L^{\frac{2 N p - 2 N + 2 p}{(N + α) p - 2 N + 2 p}} (ℝ^{N})$ , so

$\begin{array}{l} \int_{ℝ^{N}} (I_{α} * {| u_{n} - u |}^{q}) {| u |}^{q - 2} u φ \\ \leq {(\int_{ℝ^{N}} {(I_{α} * {| u_{n} - u |}^{q})}^{\frac{N p}{N p - N + p}} {| u |}^{\frac{N p (q - 1)}{N p - N + p}} d x)}^{\frac{N p - N + p}{N p}} \cdot {‖ φ ‖}_{W^{1, p} (ℝ^{N})} \\ = o_{n} (1) {‖ φ ‖}_{W^{1, p} (ℝ^{N})} . \end{array}$

In a similar way, we can verify that

$\int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u_{n} - u |}^{q - 2} (u_{n} - u) φ = o_{n} (1) {‖ φ ‖}_{W^{1, p} (ℝ^{N})} .$

Therefore, conclusion (2) is established. $■$

Let

$J_{ε} (u) = \frac{1}{p} \int_{ℝ^{N}} A (x, u) {| \nabla u |}^{p} d x + \frac{1}{p} \int_{ℝ^{N}} E (ε x) {| u |}^{p} d x + σ \int_{ℝ^{N}} K_{ε} (x, u) d x .$

The definition operator $F : X_{ε} \to X_{ε}$ , $v = F u \in X_{ε}$ is the only solution of the following equation.

$\begin{array}{l} \frac{1}{2} 〈 {J^{'}}_{ε} (u) + {J^{'}}_{ε} (v) - d J_{ε} (u - v), η 〉 + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| v |}^{p - 2} v η d x \\ = \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u)) \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u η d x, \forall η \in X_{ε} \end{array}$ (15)

Lemma 3.9 $\forall u, v \in X_{ε}$

(1) For $p \geq 2$ ,

$\begin{array}{l} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v), φ 〉 \\ \leq c ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p - 2} + {‖ v ‖}_{W^{1, p} (ℝ^{N})}^{p - 2}) {‖ u - v ‖}_{W^{1, p} (ℝ^{N})} {‖ φ ‖}_{W^{1, p} (ℝ^{N})} \\ + c ({‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2} + {‖ v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2}) {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})} {‖ φ ‖}_{L_{ε}^{m} (ℝ^{N})}, \end{array}$

For $1 < p < 2$ ,

$\begin{array}{l} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v), φ 〉 \\ \leq c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} {‖ φ ‖}_{W^{1, p} (ℝ^{N})} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1} {‖ φ ‖}_{L_{ε}^{m} (ℝ^{N})}) . \end{array}$

(2) For $p > 1$ ,

$〈 {J^{'}}_{ε} (u), φ 〉 \leq c ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} {‖ φ ‖}_{W^{1, p} (ℝ^{N})} + {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1} {‖ φ ‖}_{L_{ε}^{m} (ℝ^{N})}) .$

(3) For $p \geq 2$ ,

$〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v), u - v 〉 \geq c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}),$

For $1 < p < 2$ ,

$\begin{array}{l} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v), u - v 〉 \\ \geq c {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{2} {({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{2 - p} + {‖ v ‖}_{W^{1, p} (ℝ^{N})}^{2 - p})}^{- 1} \\ + c {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{2} {({‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{2 - m} + {‖ v ‖}_{L_{ε}^{m} (ℝ^{N})}^{2 - m})}^{- 1} . \end{array}$

Proof: (1) From $V_{1}, A_{1}$ and Hölder inequality it follows that

$\begin{array}{l} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v), φ 〉 \\ = \int_{ℝ^{N}} (A (x, u) {| \nabla u |}^{p - 2} \nabla u - A (x, v) {| \nabla v |}^{p - 2} \nabla v) \nabla φ d x \\ + \frac{1}{p} \int_{ℝ^{N}} (A_{t} (x, u) {| \nabla u |}^{p} - A_{t} (x, v) {| \nabla v |}^{p}) φ d x \\ + \int_{ℝ^{N}} E (ε x) ({| u |}^{p - 2} u - {| v |}^{p - 2} v) φ d x + σ \int_{ℝ^{N}} (k_{ε} (x, u) - k_{ε} (x, v)) φ d x \end{array}$

$\begin{array}{l} \leq c \int_{ℝ^{N}} | {| \nabla u |}^{p - 2} \nabla u - {| \nabla v |}^{p - 2} \nabla v | \cdot | \nabla φ | d x + c \int_{ℝ^{N}} | {| \nabla u |}^{p} - {| \nabla v |}^{p} | \cdot | φ | d x \\ + c \int_{ℝ^{N}} | {| u |}^{p - 2} u - {| v |}^{p - 2} v | \cdot | φ | d x + σ \int_{ℝ^{N}} (k_{ε} (x, u) - k_{ε} (x, v)) φ d x \\ \leq c {(\int_{ℝ^{N}} {| {| \nabla u |}^{p - 2} \nabla u - {| \nabla v |}^{p - 2} \nabla v |}^{\frac{p}{p - 1}} d x)}^{\frac{p - 1}{p}} {‖ φ ‖}_{W^{1, p} (ℝ^{N})} \\ + c {(\int_{ℝ^{N}} {| {| u |}^{p - 2} u - {| v |}^{p - 2} v |}^{\frac{p}{p - 1}} d x)}^{\frac{p - 1}{p}} {‖ φ ‖}_{W^{1, p} (ℝ^{N})} \\ + σ \int_{ℝ^{N}} (k_{ε} (x, u) - k_{ε} (x, v)) φ d x . \end{array}$

For $p \geq 2$ , from (9) we have

$\begin{array}{l} {(\int_{ℝ^{N}} {| {| \nabla u |}^{p - 2} \nabla u - {| \nabla v |}^{p - 2} \nabla v |}^{\frac{p}{p - 1}} d x)}^{\frac{p - 1}{p}} \\ \leq c {(\int_{ℝ^{N}} {({| \nabla u |}^{p - 2} + {| \nabla v |}^{p - 2})}^{\frac{p}{p - 1}} \cdot {| \nabla u - \nabla v |}^{\frac{p}{p - 1}} d x)}^{\frac{p - 1}{p}} \\ \leq c ({(\int_{ℝ^{N}} {| \nabla u |}^{p} d x)}^{\frac{p - 2}{p}} + {(\int_{ℝ^{N}} {| \nabla v |}^{p} d x)}^{\frac{p - 2}{p}}) \cdot {(\int_{ℝ^{N}} {| \nabla u - \nabla v |}^{p} d x)}^{\frac{1}{p}} \\ \leq c ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p - 2} + {‖ v ‖}_{W^{1, p} (ℝ^{N})}^{p - 2}) {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}, \end{array}$

similarly,

${(\int_{ℝ^{N}} {| {| u |}^{p - 2} u - {| v |}^{p - 2} v |}^{\frac{p}{p - 1}} d x)}^{\frac{p - 1}{p}} \leq c ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p - 2} + {‖ v ‖}_{W^{1, p} (ℝ^{N})}^{p - 2}) {‖ u - v ‖}_{W^{1, p} (ℝ^{N})} .$

For $1 < p < 2$ , from (10) we have

$\begin{array}{l} {(\int_{ℝ^{N}} {| {| \nabla u |}^{p - 2} \nabla u - {| \nabla v |}^{p - 2} \nabla v |}^{\frac{p}{p - 1}} d x)}^{\frac{p - 1}{p}} \\ \leq c {(\int_{ℝ^{N}} {| \nabla u - \nabla v |}^{p} d x)}^{\frac{p - 1}{p}} \leq c {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 1}, \end{array}$

${(\int_{ℝ^{N}} {| {| u |}^{p - 2} u - {| v |}^{p - 2} v |}^{\frac{p}{p - 1}} d x)}^{\frac{p - 1}{p}} \leq c {(\int_{ℝ^{N}} {| u - v |}^{p} d x)}^{\frac{p - 1}{p}} \leq c {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} .$

In addition, by Lemma 3.1 (6) and Hölder inequality, for $p \geq 2$ ,

$\begin{array}{l} σ \int_{ℝ^{N}} (k_{ε} (x, u) - k_{ε} (x, v)) φ d x \\ \leq c \int_{ℝ^{N}} \exp {(m - p) d i s t (ε x, M)} ({| u |}^{m - 2} + {| v |}^{m - 2}) | u - v | | φ | d x \\ \leq c ({‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2} + {‖ v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2}) {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})} {‖ φ ‖}_{L_{ε}^{m} (ℝ^{N})} . \end{array}$

For $1 < p < 2$ ,

$\begin{array}{l} σ \int_{ℝ^{N}} (k_{ε} (x, u) - k_{ε} (x, v)) φ d x \\ \leq c \int_{ℝ^{N}} \exp {(m - p) d i s t (ε x, M)} {| u - v |}^{m - 1} | φ | d x \\ \leq c {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1} {‖ φ ‖}_{L_{ε}^{m} (ℝ^{N})} . \end{array}$

(2) Take $v = 0$ in (1) to get (2).

(3) From assume A₁, A₂, we have

$\begin{array}{l} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v), u - v 〉 \\ = \int_{ℝ^{N}} (A (x, u) {| \nabla u |}^{p - 2} \nabla u - A (x, v) {| \nabla v |}^{p - 2} \nabla v) (\nabla u - \nabla v) d x \\ + \frac{1}{p} \int_{ℝ^{N}} (A_{t} (x, u) {| \nabla u |}^{p} - A_{t} (x, v) {| \nabla v |}^{p}) (u - v) d x \\ + \int_{ℝ^{N}} E (ε x) ({| u |}^{p - 2} u - {| v |}^{p - 2} v) (u - v) d x \\ + σ \int_{ℝ^{N}} (k_{ε} (x, u) - k_{ε} (x, v)) (u - v) d x \end{array}$

$\begin{array}{l} \geq c_{1} \int_{ℝ^{N}} ({| \nabla u |}^{p - 2} \nabla u - {| \nabla v |}^{p - 2} \nabla v) (\nabla u - \nabla v) d x \\ + c \int_{ℝ^{N}} ({| u |}^{p - 2} u - {| v |}^{p - 2} v) (u - v) d x \\ + σ \int_{ℝ^{N}} (k_{ε} (x, u) - k_{ε} (x, v)) (u - v) d x . \end{array}$

For $p \geq 2$ , from (9) we have

$\begin{array}{l} \int_{ℝ^{N}} ({| \nabla u |}^{p - 2} \nabla u - {| \nabla v |}^{p - 2} \nabla v) (\nabla u - \nabla v) d x \\ \geq c \int_{ℝ^{N}} {| \nabla u - \nabla v |}^{p} d x \geq c {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p}, \end{array}$

$\int_{ℝ^{N}} ({| u |}^{p - 2} u - {| v |}^{p - 2} v) (u - v) d x \geq c {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p} .$

From Lemma 3.1 (5), we have

$\begin{array}{l} \int_{ℝ^{N}} (k_{ε} (x, u) - k_{ε} (x, v)) (u - v) d x \\ \geq c \int_{ℝ^{N}} \exp {(m - p) d i s t (ε x, M)} {| u - v |}^{m} d x = c {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m} . \end{array}$

Similarly, from (10) and Lemma 3.1 (5) can prove the case of $1 < p < 2$ , so (3) holds. $■$

Lemma 3.10 If ${‖ u ‖}_{X_{ε}}$ is bounded, then ${‖ F u ‖}_{X_{ε}} = {‖ v ‖}_{X_{ε}}$ is bounded.

Proof: By (15)

$\begin{array}{l} {‖ F u ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ F u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m} \\ \leq c 〈 {J^{'}}_{ε} (v), v 〉 \leq c h_{λ} (ψ^{\frac{1}{2}} (u)) \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u v d x \\ \leq c {‖ u ‖}_{W^{1, p} (ℝ^{N})}^{2 q - 1} {‖ F u ‖}_{W^{1, p} (ℝ^{N})} \leq c {‖ F u ‖}_{W^{1, p} ( ℝ N )} \end{array}$

thus ${‖ F u ‖}_{X_{ε}}$ is bounded. $■$

Lemma 3.11 F is odd, well defined, and continuous on $X_{ε}$ .

Proof: From the definition of operator F, it is easy to know that F is an odd operator. Definition

$\begin{array}{l} G (v) = \frac{1}{2} J_{ε} (v) + \frac{1}{2} J_{ε} (v - u) + \frac{1}{2} 〈 {J^{'}}_{ε} (u), v 〉 \\ + \frac{1}{p} {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| v |}^{p} d x \\ - \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u)) \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u v d x, \forall v \in X_{ε} \end{array}$

Equation (15) has a unique solution $v = F u$ , which can be obtained by solving the minimization problem $\inf {G (v) | v \in X_{ε}}$ . Since

$G (v) \geq \frac{1}{2} J_{ε} (v) + \frac{1}{2} 〈 {J^{'}}_{ε} (u), v 〉 - c_{λ} \geq c_{1} {‖ v ‖}_{W^{1, p} (ℝ^{N})}^{p} - c_{2} {‖ v ‖}_{W^{1, p} (ℝ^{N})} - c_{λ}$

thus G is coercive.

Let ${v_{n}} \subset X_{ε}$ be a minimizing sequence for the functional G, $v_{n} \to v$ in $X_{ε}$ . By the lower semicontinuity

$G (v) \leq \underset{n \to \infty}{\lim \inf} G (v_{n}) = \inf {G (v) | v \in X_{ε}}$

so v is a solution of (15). Assume $v_{1}, v_{2}$ are solutions of (15), taking $v_{1} - v_{2}$ as the test function, we have

$\begin{array}{l} \frac{1}{2} 〈 {J^{'}}_{ε} (v_{1}) - {J^{'}}_{ε} (v_{2}), v_{1} - v_{2} 〉 + \frac{1}{2} 〈 {J^{'}}_{ε} (v_{1} - u) - {J^{'}}_{ε} (v_{2} - u), v_{1} - v_{2} 〉 \\ + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) ({| v_{1} |}^{p - 2} v_{1} - {| v_{2} |}^{p - 2} v_{2}) (v_{1} - v_{2}) d x = 0 \end{array}$

Hence

$〈 {J^{'}}_{ε} (v_{1}) - {J^{'}}_{ε} (v_{2}), v_{1} - v_{2} 〉 = 0.$

By Lemma 3.9 (3), for $p \geq 2$

$〈 {J^{'}}_{ε} (v_{1}) - {J^{'}}_{ε} (v_{2}), v_{1} - v_{2} 〉 \geq c ({‖ v_{1} - v_{2} ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ v_{1} - v_{2} ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}),$

for $1 < p < 2$

$\begin{array}{l} 〈 {J^{'}}_{ε} (v_{1}) - {J^{'}}_{ε} (v_{2}), v_{1} - v_{2} 〉 \\ \geq c {‖ v_{1} - v_{2} ‖}_{W^{1, p} (ℝ^{N})}^{2} {({‖ v_{1} ‖}_{W^{1, p} (ℝ^{N})}^{2 - p} + {‖ v_{2} ‖}_{W^{1, p} (ℝ^{N})}^{2 - p})}^{- 1} \\ + c {‖ v_{1} - v_{2} ‖}_{L_{ε}^{m} (ℝ^{N})}^{2} {({‖ v_{1} ‖}_{L_{ε}^{m} (ℝ^{N})}^{2 - m} + {‖ v_{2} ‖}_{L_{ε}^{m} (ℝ^{N})}^{2 - m})}^{- 1} . \end{array}$

Then $v_{1} = v_{2}$ , we prove that Equation (15) has a unique solution of $v = F u$ .

The following proves that F is continuous, taking $η = v_{n} - v$ in (15), we have

$\begin{array}{l} \frac{1}{2} 〈 {J^{'}}_{ε} (u - v) - {J^{'}}_{ε} (u_{n} - v_{n}), (u - v) - (u_{n} - v_{n}) 〉 + \frac{1}{2} 〈 {J^{'}}_{ε} (v_{n}) - {J^{'}}_{ε} (v), v_{n} - v 〉 \\ + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) ({| v_{n} |}^{p - 2} v_{n} - {| v |}^{p - 2} v) (v_{n} - v) d x \\ = \frac{1}{2} 〈 {J^{'}}_{ε} (u - v) - {J^{'}}_{ε} (u_{n} - v_{n}), u - u_{n} 〉 + \frac{1}{2} 〈 {J^{'}}_{ε} (u_{n}) - {J^{'}}_{ε} (u), v - v_{n} 〉 \\ + [{(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β - 1} - {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1}] \int_{ℝ^{N}} χ_{ε} (x) {| v |}^{p - 2} v (v - v_{n}) d x \\ + \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u_{n})) \int_{ℝ^{N}} ((I_{α} * {| u_{n} |}^{q}) {| u_{n} |}^{q - 2} u_{n} - (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u) (v_{n} - v) d x \\ + \frac{1}{2} (h_{λ} (ψ^{\frac{1}{2}} (u_{n})) - h_{λ} (ψ^{\frac{1}{2}} (u))) \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u (v_{n} - v) d x . \end{array}$ (16)

Suppose $u_{n} \to u$ in $X_{ε}$ , by Lemma 3.9 and Lemma 3.10, for $p \geq 2$ , we have

$\begin{array}{l} 〈 {J^{'}}_{ε} (u - v) - {J^{'}}_{ε} (u_{n} - v_{n}), u - u_{n} 〉 + 〈 {J^{'}}_{ε} (u_{n}) - {J^{'}}_{ε} (u), v - v_{n} 〉 \\ \leq c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 2} + {‖ u_{n} - v_{n} ‖}_{W^{1, p} (ℝ^{N})}^{p - 2}) {‖ u - v - u_{n} + v_{n} ‖}_{W^{1, p} (ℝ^{N})} {‖ u - u_{n} ‖}_{W^{1, p} (ℝ^{N})} \\ + c ({‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2} + {‖ u_{n} - v_{n} ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2}) {‖ u - v - u_{n} + v_{n} ‖}_{L_{ε}^{m} (ℝ^{N})} {‖ u - u_{n} ‖}_{L_{ε}^{m} (ℝ^{N})} \\ + c ({‖ u_{n} ‖}_{W^{1, p} (ℝ^{N})}^{p - 2} + {‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p - 2}) {‖ u_{n} - u ‖}_{W^{1, p} (ℝ^{N})} {‖ v - v_{n} ‖}_{W^{1, p} (ℝ^{N})} \\ + c ({‖ u_{n} ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2} + {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2}) {‖ u_{n} - u ‖}_{L_{ε}^{m} (ℝ^{N})} {‖ v - v_{n} ‖}_{L_{ε}^{m} (ℝ^{N})} \\ \leq c {‖ u_{n} - u ‖}_{X_{ε}} = o_{n} (1), \end{array}$ (17)

Similarly, for $1 < p < 2$ , we have

$\begin{array}{l} 〈 {J^{'}}_{ε} (u - v) - {J^{'}}_{ε} (u_{n} - v_{n}), u - u_{n} 〉 + 〈 {J^{'}}_{ε} (u_{n}) - {J^{'}}_{ε} (u), v - v_{n} 〉 \\ \leq c ({‖ u - v - u_{n} + v_{n} ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} {‖ u - u_{n} ‖}_{W^{1, p} (ℝ^{N})} + {‖ u - v - u_{n} + v_{n} ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1} {‖ u - u_{n} ‖}_{L_{ε}^{m} (ℝ^{N})}) \\ + c ({‖ u_{n} - u ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} {‖ v - v_{n} ‖}_{W^{1, p} (ℝ^{N})} + {‖ u_{n} - u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1} {‖ v - v_{n} ‖}_{L_{ε}^{m} (ℝ^{N})}) \\ = o_{n} (1) . \end{array}$ (18)

By Lemma 3.10 and Hardy-Littlewood-Sobolev inequality, we obtain that

$\begin{array}{l} [{(\int_{ℝ^{N}} χ_{ε} (x) {| u_{n} |}^{p} d x - 1)}_{+}^{β - 1} - {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1}] \int_{ℝ^{N}} χ_{ε} (x) {| v |}^{p - 2} v (v - v_{n}) d x \\ + \frac{1}{2} (h_{λ} (ψ^{\frac{1}{2}} (u_{n})) - (ψ^{\frac{1}{2}} (u))) \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u (v_{n} - v) d x = o_{n} (1) . \end{array}$ (19)

By Lemma 3.5 and Lemma 3.10, we have

$\begin{array}{l} \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u_{n})) \int_{ℝ^{N}} ((I_{α} * {| u_{n} |}^{q}) {| u_{n} |}^{q - 2} u_{n} - (I_{α} * {| u |}^{q}) {| u |}^{q - 2} u) (v_{n} - v) d x \\ = \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u_{n})) \int_{ℝ^{N}} (I_{α} * {| u_{n} - u |}^{q}) {| u_{n} - u |}^{q - 2} (u_{n} - u) (v_{n} - v) d x \\ = o_{n} (1) . \end{array}$ (20)

Thus the right-hand side of (16) satisfies

$R H S = o_{n} (1) .$ (21)

Next, we estimate the left-hand side of (16), for $p \geq 2$ , by (9)

$\begin{matrix} L H S \geq 〈 {J^{'}}_{ε} (v_{n}) - {J^{'}}_{ε} (v), v_{n} - v 〉 \\ \geq c ({‖ v_{n} - v ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ v_{n} - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}), \end{matrix}$ (22)

for $1 < p < 2$ , by (10)

$\begin{matrix} L H S \geq 〈 {J^{'}}_{ε} (v_{n}) - {J^{'}}_{ε} (v), v_{n} - v 〉 \\ \geq c {‖ v_{n} - v ‖}_{W^{1, p} (ℝ^{N})}^{2} {({‖ v_{n} ‖}_{W^{1, p} (ℝ^{N})}^{2 - p} + {‖ v ‖}_{W^{1, p} (ℝ^{N})}^{2 - p})}^{- 1} \\ + {‖ v_{n} - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{2} {({‖ v_{n} ‖}_{L_{ε}^{m} (ℝ^{N})}^{2 - m} + {‖ v ‖}_{L_{ε}^{m} (ℝ^{N})}^{2 - m})}^{- 1} . \end{matrix}$ (23)

From (21) to (23), for $p > 1$ , we obtain

${‖ v_{n} - v ‖}_{X_{ε}} \to 0, as n \to \infty .$

Therefore, F is continuous. $■$

We verify the condition (F₁) in Theorem 2.2

Lemma 3.12 If $u \in X_{ε}, v = F u$ , then

(1) $〈 {Γ^{'}}_{ε, λ} (u), u - v 〉 \geq c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m})$

(2) $\begin{array}{l} ‖ {Γ^{'}}_{ε, λ} (u) ‖ \leq c {(1 + | Γ_{ε, λ} (u) | + {‖ u - v ‖}_{X_{ε}})}^{α} {‖ u - v ‖}_{X_{ε}}, p \geq 2 \\ ‖ {Γ^{'}}_{ε, λ} (u) ‖ \leq c {‖ u - v ‖}_{X_{ε}}^{α - 1}, 1 < p < 2 \end{array}$

where $α = \max {p, m, p β}$

Proof: (1) By (15), $\forall η \in X_{ε}$ we have

$\begin{array}{l} 〈 {Γ^{'}}_{ε, λ} (u), η 〉 \\ = 〈 {J^{'}}_{ε} (u), η 〉 + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p - 2} u η d x \\ - \frac{1}{2} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v) + {J^{'}}_{ε} (v - u), η 〉 \\ - {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| v |}^{p - 2} v η d x \\ = \frac{1}{2} 〈 {J^{'}}_{ε} (u - v), u - v 〉 + \frac{1}{2} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v), u - v 〉 \\ + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) ({| u |}^{p - 2} u - {| v |}^{p - 2} v) η d x . \end{array}$ (24)

Hence

$\begin{array}{l} 〈 {Γ^{'}}_{ε, λ} (u), u - v 〉 \\ = \frac{1}{2} 〈 {J^{'}}_{ε} (u - v), u - v 〉 + \frac{1}{2} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v), u - v 〉 \\ + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) ({| u |}^{p - 2} u - {| v |}^{p - 2} v) (u - v) d x \\ \geq c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}) . \end{array}$

(2) For $p \geq 2$ , by (7) (15) we have

$\begin{array}{l} μ Γ_{ε, λ} (u) - \frac{1}{2} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v) + {J^{'}}_{ε} (u - v), u 〉 \\ = μ Γ_{ε, λ} (u) - \frac{1}{q} 〈 {J^{'}}_{ε} (u), u 〉 + \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u)) \int_{ℝ^{N}} (I_{α} * {| u |}^{q}) {| u |}^{q} d x \\ - {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| v |}^{p - 2} v u d x \\ = \frac{1}{p} \int_{ℝ^{N}} ((μ - p) A (x, u) - A_{t} (x, u)) {| \nabla u |}^{p} d x + \frac{μ - p}{p} \int_{ℝ^{N}} E (ε x) {| u |}^{p} d x \\ + σ μ \int_{ℝ^{N}} K_{ε} (x, u) d x - σ \int_{ℝ^{N}} k_{ε} (x, u) u + \frac{μ}{p β} {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} \end{array}$

$\begin{array}{l} - {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| v |}^{p - 2} v u d x \\ + \frac{1}{2} h_{λ} (ψ^{\frac{1}{2}} (u)) ψ (u) d x - \frac{μ}{2 q} g_{λ} (ψ^{\frac{1}{2}} (u)) ψ (u) d x \\ \geq c {‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p} + \frac{σ (μ - m)}{m} \int_{ℝ^{N}} k_{ε} (x, u) u d x + \frac{μ}{p β} {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} \\ - {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| v |}^{p - 2} v u d x, \end{array}$

while

$\begin{array}{l} \frac{σ (μ - m)}{m} \int_{ℝ^{N}} k_{ε} (x, u) u d x \\ \geq c \int_{ℝ^{N}} \exp {(m - p) d i s t (ε x, M)} {| u |}^{m} d x \geq c {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m} . \end{array}$

By Höder inequality and Young inequality, we obtain that

$\begin{array}{l} \frac{μ}{p β} {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} - {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| v |}^{p - 2} v u d x \\ = \frac{μ}{p β} {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} - {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x \\ + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) ({| u |}^{p - 2} u - {| v |}^{p - 2} v) u d x \\ \geq c {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} - c {(\int_{ℝ^{N}} χ_{ε} (x) ({| u |}^{p - 2} + {| v |}^{p - 2}) | u - v | | u | d x)}^{β} - c \\ \geq c {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} - c {(\int_{ℝ^{N}} χ_{ε} (x) {| u - v |}^{p} d x)}^{β} - c, \end{array}$

Hence

$\begin{array}{l} μ Γ_{ε, λ} (u) - \frac{1}{2} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v) + {J^{'}}_{ε} (u - v), u 〉 \\ \geq c ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}) + c {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} - c {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p β} - c . \end{array}$ (25)

In addition, by Lemma 3.9

$\begin{array}{l} μ Γ_{ε, λ} (u) - \frac{1}{2} 〈 {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v) + {J^{'}}_{ε} (u - v), u 〉 \\ \leq c | Γ_{ε, λ} (u) | + c ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p - 2} + {‖ v ‖}_{W^{1, p} (ℝ^{N})}^{p - 2}) {‖ u - v ‖}_{W^{1, p} (ℝ^{N})} {‖ u ‖}_{W^{1, p} (ℝ^{N})} \\ + c ({‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2} + {‖ v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2}) {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})} {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})} \\ + c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} {‖ u ‖}_{W^{1, p} (ℝ^{N})} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1} {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}) \end{array}$

$\begin{array}{l} \leq c | Γ_{ε, λ} (u) | + c {‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} {‖ u - v ‖}_{W^{1, p} (ℝ^{N})} + c {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1} {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})} \\ + c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} {‖ u ‖}_{W^{1, p} (ℝ^{N})} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1} {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}) \\ \leq c | Γ_{ε, λ} (u) | + c ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}) + c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}), \end{array}$ (26)

that is

$\begin{array}{l} c ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}) + c {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} - c {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p β} - c \\ \leq c | Γ_{ε, λ} (u) | + c ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}) + c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m}) . \end{array}$

Thus

$\begin{array}{l} {‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m} + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β} \\ \leq c (1 + | Γ_{ε, λ} (u) | + {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m} + {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p β}) . \end{array}$ (27)

By (24) and Lemma 3.9(1), (27) and Young inequality, we have

$\begin{array}{l} ‖ {Γ^{'}}_{ε, λ} (u) ‖ \leq \frac{1}{2} ‖ {J^{'}}_{ε} (u - v) ‖ + \frac{1}{2} ‖ {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v) ‖ \\ + | {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) ({| u |}^{p - 2} u - {| v |}^{p - 2} v) d x | \\ \leq c (1 + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1}) ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p - 2} + {‖ v ‖}_{W^{1, p} (ℝ^{N})}^{p - 2}) {‖ u - v ‖}_{W^{1, p} (ℝ^{N})} \\ + c ({‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2} + {‖ v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2}) {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})} + c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1}) \end{array}$

$\begin{array}{l} \leq c (1 + {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1}) ({‖ u ‖}_{W^{1, p} (ℝ^{N})}^{p - 2} + {‖ v ‖}_{W^{1, p} (ℝ^{N})}^{p - 2}) {‖ u - v ‖}_{X_{ε}} \\ + c ({‖ u ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2} + {‖ v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 2}) {‖ u - v ‖}_{X_{ε}} \\ \leq c {(1 + | Γ_{ε, λ} (u) | + {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m} + {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p β})}^{2} {‖ u - v ‖}_{X_{ε}} \\ \leq c {(1 + | Γ_{ε, λ} (u) | + {‖ u - v ‖}_{X_{ε}})}^{α} {‖ u - v ‖}_{X_{ε}} . \end{array}$

where $α = \max {p β, p, m}$ .

For $1 < p < 2$ , by (24) and Lemma 3.9 (2), we have

$\begin{array}{l} ‖ {Γ^{'}}_{ε, λ} (u) ‖ \leq \frac{1}{2} ‖ {J^{'}}_{ε} (u - v) ‖ + \frac{1}{2} ‖ {J^{'}}_{ε} (u) - {J^{'}}_{ε} (v) ‖ \\ + | {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \int_{ℝ^{N}} χ_{ε} (x) ({| u |}^{p - 2} u - {| v |}^{p - 2} v) d x | \\ \leq c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1}) + c {‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} {(\int_{ℝ^{N}} χ_{ε} (x) {| u |}^{p} d x - 1)}_{+}^{β - 1} \\ \leq c ({‖ u - v ‖}_{W^{1, p} (ℝ^{N})}^{p - 1} + {‖ u - v ‖}_{L_{ε}^{m} (ℝ^{N})}^{m - 1}) \\ \leq c {‖ u - v ‖}_{X_{ε}}^{α - 1} . \end{array}$

$■$

From Lemma 3.12, we can get the following corollary:

Corollary 3.13 For all $b_{0}, c_{0} > 0$ , there exists $b = b (b_{0}, c_{0}) > 0$ such that if

$| Γ_{ε, λ} (u) | \leq c_{0}, ‖ {Γ^{'}}_{ε, λ} (u) ‖ \geq b_{0}$

then

$〈 {Γ^{'}}_{ε, λ} (u), u - F u 〉 \geq b {‖ u - F u ‖}_{X_{ε}} > 0, u - F u \neq 0.$

we denote

$D (f, g) = \int_{ℝ^{N}} \int_{ℝ^{N}} \frac{f (x) g (y)}{{| x - y |}^{N - α}} d x d y$

Lemma 3.14 ( [5] , Theorem 9.8) If $N \geq 3, 0 < α < N$ , and $D (f, f), D (g, g) < \infty$ , then

${| D (f, g) |}^{2} \leq D (f, f) \cdot D (g$

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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