Existence and Asymptotic Behavior of Localized Nodal Solutions for a Class of Kirchhoff-Type Equations

Abstract

In this paper, we study the existence and asymptotic behavior of localized nodal solutions for the following Kirchhoff-type equation

$$\begin{aligned} -\left( \varepsilon ^2a+\varepsilon b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(x)u=|u|^{p-2}u, \ x \in \mathbb {R}^3, \end{aligned}$$

where $a>0$, $b>0$ and $4<p<6$. Under only a local condition that V has a local trapping potential well, when $\varepsilon >0$ is sufficiently small, we construct the existence of a sequence of localized nodal solutions concentrating around the local minimum points of the potential function V by using variational method and penalization approach. Moreover, we regard b as a parameter and study the asymptotic behavior of the nodal solutions as $b\searrow 0$, which reflects some relationship between $b>0$ and $b=0$.

1 Introduction and Main Results

Consider the following Kirchhoff-type equation

$$\begin{aligned} -\left( \varepsilon ^2a+\varepsilon b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(x)u=|u|^{p-2}u, \ x \in \mathbb {R}^3, \end{aligned}$$

(1.1)

where $\varepsilon >0$, $a>0$, $b>0$ and $4<p<6$. The interest for studying Kirchhoff type equations is twofold: firstly, the interest comes from the physical background of Kirchhoff type equations. Indeed, if we set $\varepsilon =1$, $V(x)=0$, a bounded domain $\Omega \subset \mathbb {R}^3$ and a new nonlinearity g(x, u) instead of $\mathbb {R}^3$ and $|u|^{p-2}u$ respectively, it reduces to the following Kirchhoff Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{l} -\left( a+b\int _{\Omega }|\nabla u|^2dx\right) \Delta u=g(x,u), \ \hbox {in} \ \Omega ,\\ u=0, \ \hbox {on} \ \partial \Omega ,\\ \end{array} \right. \end{aligned}$$

(1.2)

which is related to the stationary analogue of the equation

$$\begin{aligned} \rho \frac{\partial ^2u}{\partial t^2}-\left( \frac{P_0}{h}+\frac{E}{2L}\int _0^L|\frac{\partial u}{\partial x}|^2dx\right) \frac{\partial ^2u}{\partial x^2}=0, \end{aligned}$$

where u denotes the displacement, g(x, u) the external force and b the initial tension while a is related to the intrinsic properties of the string, such as Young’ modulus. This equation has been introduced for the first time in 1883 by Kirchhoff [25] in dimension 1, without forcing term and with Dirichlet boundary conditions, to extend the classical D’ Alembert’ s wave equations for free vibration of elastic string. After the pioneering work by Lions [31], in which a functional analysis approach was proposed to the equation

$$\begin{aligned} \left\{ \begin{array}{l} u_{tt}-\left( a+b\int _{\Omega }|\nabla u|^2dx\right) \Delta u=g(x,u), \ x \in \Omega ,\\ u=0, \ x \in \partial \Omega ,\\ \end{array} \right. \end{aligned}$$

there is a considerable amount of work on investigating the properties of this type equations, especially

$$\begin{aligned} \left\{ \begin{array}{l} -\left( a+b\int _{\Omega }|\nabla u|^2dx\right) \Delta u+V(x)u=g(x,u), \ x \in \Omega ,\\ u \in H_0^1(\Omega ).\\ \end{array} \right. \end{aligned}$$

Here, $\Omega$ is a domain in $\mathbb {R}^3$, possibly unbounded, with empty or smooth boundary, $V: \Omega \rightarrow \mathbb {R}$, $g \in C(\Omega \times \mathbb {R}, \mathbb {R})$.

Second, the term $(\int _{\Omega }|\nabla u|^2dx) \Delta u$ depends not only on the pointwise value of $\Delta u$, but also on the integral of $|\nabla u|^2$ over the domain $\Omega$. Hence, Eq. (1.2) is often referred to as being nonlocal due to the appearance of the term $(\int _{\Omega }|\nabla u|^2dx) \Delta u$ and this fact indicates that Eq. (1.2) is not a usual pointwise equality and is very different from classical elliptic equation. The competing effect of the nonlocal term with the nonlinearity and the lack of compactness of the embedding of $H^1(\mathbb {R}^3)$ into the space $L^t(\mathbb {R}^3)$, $t \in (2,6)$, prevents us from using the variational method in a standard way. We point out that existence, multiplicity and concentration of solutions for degenerate case involving subcritical, critical and supercritical growth have been extensively studied under different assumptions about the potential V. Via invariant sets of descent flow, sign-changing solutions of (1.2) were established in [35, 49] on the bounded domain. Combining constraint minimization method and quantitative deformation lemma, Shuai [40] studied the existence of least energy sign-changing solutions of (1.2). Later, this result has been extended to more general nonlinearity by Tang and Cheng [43]. But they all do not consider the potential function V(x).

A solution u(x) is referred to as a bound state of (1.1) if $u(x)\rightarrow 0$ as $|x|\rightarrow +\infty$. When $\varepsilon >0$ is sufficiently small, bound states of (1.1) are called semiclassical states and an important feature of semiclassical states is their concentration as $\varepsilon \rightarrow 0$. He and Zou [19] seems to be the first to study the singularly perturbed Kirchhoff equation (1.1). Under the global condition

$$\begin{aligned} 0<V_0:=\inf \limits _{x \in \mathbb {R}^3}V(x)<\liminf \limits _{|x|\rightarrow \infty }V(x), \end{aligned}$$

(1.3)

using of variational method, they showed that the multiplicity of positive solutions concentrating on global minima of V(x) as $\varepsilon \rightarrow 0$. Later, under the same global condition, Wang et al. [45] extended the results of [19] to the critical growth case. Recently, when V satisfies (1.3) and nonlinearity is of super-linear growth at infinity and satisfies neither the usual Ambrosetti–Rabinowitz type condition nor monotonicity condition, Lin and Wei [30] obtained a ground state solution concentrating around global minimum of V for Kirchhoff type equation like (1.1). When V only satisfies local condition, i.e., there exists a bounded open set $\Omega \subset \mathbb {R}^3$ such that

$$\begin{aligned} \inf \limits _{x\in \Omega }V(x)<\inf \limits _{x\in \partial \Omega }V(x), \end{aligned}$$

He et al. [18] constructed a semiclassical solution $u_{\varepsilon }$ concentrating around a local minimum point of V in $\Omega$ by using penalization method. Moreover, combining minimax theorems and Ljusternik–Schnirelmann theory, they also obtained multiple solutions by employing the topology construct of the set where the potential V attains its minimum. This work can be viewed as extension of [45] from global case to local case. Subsequently, they [17] showed that the existence of positive solutions for the following equation

$$\begin{aligned} -\left( \varepsilon ^2a+\varepsilon b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(x)u=u^q+u^5, \ u>0, \ x \in \mathbb {R}^3 \end{aligned}$$

for $\varepsilon >0$ sufficiently small, where $1<q<3$. Li et al. [28] established the existence and local uniqueness of solutions to Eq. (1.1) using local Pohozaev identity and Lyapunov–Schmidt reduction method. Recently, the authors [29] studied the concentration phenomenon of solutions for the critical Kirchhoff-type equation with competing potentials

$$\begin{aligned} -\left( \varepsilon ^2a+\varepsilon b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(x)u=P(x)f(u)+Q(x)|u|^4u, \ x \ \in \mathbb {R}^3. \end{aligned}$$

Ji and Rădulescu [20, 21] investigated the multiplicity and concentration of solutions for Kirchhoff equations with magnetic field and critical growth. Moreover, the papers [46,47,48] considered the fractional Kirchhoff problems and obtained some existence results of nontrivial solutions. More related results and recent developments, such as ground state solution, sign-changing solutions and regularity theory, we refer the readers to the papers [8, 11, 22, 26, 27, 36, 41, 42, 50] and their references therein.

Furthermore, from the mathematical point of view, Eq. (1.1) is closely related to Schrödinger equation

$$\begin{aligned} -\varepsilon ^2 \Delta u+V(x)u=g(u), \ x \in \mathbb {R}^N, \end{aligned}$$

(1.4)

when the constant b vanishes and $a=1$. The motivation of the present paper mainly comes from the similar results for Eq. (1.4), whose studies of semiclassical states goes back to the pioneer work [16] by Floer and Weinstein. Since then, it has been studied extensively under various hypotheses on the potential and the nonlinearity. Particularly, under the global condition (1.3), by using of the Mountain–Pass theorem, Rabinowitz [39] obtained a positive ground solution of (1.4) for $\varepsilon >0$ small. We point out that ground state solutions have often a special interest. They are important both for a physical and mathematical point of view since they share further properties, like stability, positivity and symmetry. Later, Wang [44] proved that the positive ground state solution obtained [39] concentrates at global minimum points of the potential function V(x). When V possesses a local minimum point $x_0$, by introducing a penalization approach, del Pino and Felmer [12] proved a single spike positive solution of (1.4) concentrates around minimizer of V in $\Lambda$, where $\Lambda$ is a bounded domain compactly contained in $\mathbb {R}^N$. The positive solution is obtained as perturbations of mountain pass positive solution of the limiting equation. By virtue of Lyapunov–Schmidt reduction method, the existence of single bump and multi-bumps solutions of (1.4) concentrating at each given nondegenerate critical point of V was showed in [37] and [38], respectively. Using the same method, Kang and Wei [23] used the mountain pass type solutions as building blocks to obtain positive solutions with multi-peaks concentrating near a saddle point $x_0$ of V. But they proved surprisingly that there could not be multi-spiked positive solutions concentrating near a local minimum point. As a result, in order to have other localized solutions near a local minimum point, we have to look for sign-changing solutions with concentrations.

In recent years, some scholars obtained a finite number of localized sign-changing solutions by various different methods. For example, Alves et al. [1] used Nehari method to study the existence of one sign-changing solution. With the aid of a dynamical systems point of view applied to the negative gradient flow in the working space of the energy functional, Bartsch et al. [5] investigated the existence of N pairs of localized sign-changing solutions concentrating near as isolated local minimum point of V for $\varepsilon >0$ small. In the light of Lyapunov–Schmidt reduction method, D’ Aprile and Pistoia [13] without assuming any symmetry conditions on the potential function V showed that there exist 9 pairs of different localized sign-changing solutions for $N\ge 3$ and 8 pairs of localized sign-changing solutions for $N=2$. But this approach is very sensitive to the numbers of positive and negative peaks. As mentioned in [14], whether there exists infinitely many localized sign-changing solutions clustered at a given local minimum point has remained large open. Furthermore, it is well known to us that by Lyapunov–Schmidt reduction method or other gluing methods, we can obtain localized sign-changing solutions, but we do not know that solutions of higher topological type (in the sense that critical points obtained by the symmetric mountain pass theorem using higher dimensional symmetric liking structures) can be localized as concentrating solutions near a local minimum point of V.

Recently, Chen and Wang [9] gave a positive answer. Specifically, without using any non-degeneracy conditions, they established multiple localized sign-changing solutions of higher topological type near a given minimum critical set of V to the semiclassical Schrödinger equation

$$\begin{aligned} -\varepsilon ^2 \Delta u+V(x)u=|u|^{p-2}u, \ x \in \mathbb {R}^N. \end{aligned}$$

Later, they [10] extended this result to critical case. Very recently, the authors in [34] extended the results in [9] to the case of quasilinear equations including the modified nonlinear Schrödinger equations

$$\begin{aligned} -\varepsilon ^2 \Delta u+V(x)u-\frac{1}{2}\varepsilon ^2 u\Delta (u^2)=|u|^{q-2}u, \ x \in \mathbb {R}^N. \end{aligned}$$

For every integer $k\ge 0$, the authors in [2] and [7] independently proved that, there is a pair of solutions $u_k^{\pm }$ having exact k nodes of

$$\begin{aligned} -\Delta u+V(|x|)u=f(|x|,u), \ x \ \in \mathbb {R}^N. \end{aligned}$$

Such solutions of the above equation are obtained by gluing solutions of the equation in each annulus, including every ball and the complement of it. However, this approach cannot be applied directly to problems with nonlocal terms, because nonlocal terms need the global information of u. This difficulty was overcome by regarding the problem as a system of $k+1$ equations with $k+1$ unknown functions $u_i$, each $u_i$ is supported on only one annulus and vanishes at the complement of it. By constructing a functional $E_k$ and a Nehari type manifold $\mathcal {N}_k$, people can find a minimizer of $E_k$ constraint on $\mathcal {N}_k$. In this way, Kim et al. [24] found infinitely many nodal solutions for Schrödinger–Poisson system, and then Deng et al. [15] treated Kirchhoff problems in $\mathbb {R}^3$. Specifically, when radial potential function V(|x|) is bounded from below by a constant $V_0>0$, they studied the existence and asymptotic behavior of nodal solutions $u_k$, which changes sign exactly k-times.

From the commentaries above, the existing work mainly focused on the existence and concentration of localized nodal solutions of Schrödinger equations. A natural question is whether we can obtain some similar results for nonlocal Kirchhoff problem like Eq. (1.1)? To the best of our best knowledge, it seems that there is almost no work on the concentration of localized nodal solutions for Eq. (1.1) under the local condition of the potential. In this paper, we shall give some answers about this topic. Motivated by the works aforementioned, under only a local condition that V has a local trapping potential well, the purpose of this paper is to establish multiple localized nodal solutions concentrating in the trapping region of the potential function V for the Kirchhoff-type equation (1.1). Compared with the loacl problems considered in [9] and [34], the problem (1.1) becomes more complicated and interesting. This is because the nonlocal term brings some new mathematical difficulties.

Before stating our results, we make the following assumptions on the potential V:

$(V_1)$ $V \in C^1(\mathbb {R}^3,\mathbb {R})$ and there exist $0<\alpha _1<\alpha _2$ such that $\alpha _1 \le V(x) \le \alpha _2$ for all $x \in \mathbb {R}^3$.

$(V_2)$ There exists a bounded domain $\mathcal {D} \subset \mathbb {R}^3$ with smooth boundary $\partial \mathcal {D}$ such that

$$\begin{aligned} \vec {n}(x) \cdot \nabla V(x)>0 \end{aligned}$$

for all $x \in \partial \mathcal {D}$, where $\vec {n}(x)$ denotes the outward normal to $\partial \mathcal {D}$ at x.

We point out that if V possesses an isolated local minimum set, i.e., V admits a local trapping potential well, then $(V_2)$ holds. By $(V_2)$, the critical set

$$\begin{aligned} \mathcal {A}:=\{x \in \mathcal {D}: \nabla V(x)=0\} \not = \emptyset \end{aligned}$$

and $\mathcal {A} \subset \mathcal {D}$ is compact. Without loss of generality, we may assume that $0 \in \mathcal {A}$. Let $B\subset \mathbb {R}^3$ and $\delta >0$, set

$$\begin{aligned} B^\delta :=\{x \in \mathbb {R}^3: \hbox {dist}(x,B)=\inf \limits _{y\in B}|x-y|<\delta \}~\hbox {and}~ B_\delta :=\{x\in \mathbb {R}^3: \delta x \in B\}. \end{aligned}$$

We are now in a position to state the main results of this paper.

Theorem 1.1

Suppose that $(V_1)$ and $(V_2)$ are satisfied. Then for any positive integer k, there exists $\varepsilon _k>0$ such that if $0<\varepsilon <\varepsilon _k$, Eq. (1.1) possesses at least k pairs of sign-changing solutions $\pm u_{j,\varepsilon }$, $j=1,2,\ldots ,k$. Moreover, for any $\delta >0$, there exist $C_1=C_1(\delta ,k)>0$ and $C_2=C_2(\delta ,k)>0$ and $\varepsilon _k(\delta )>0$ such that for $0<\varepsilon <\varepsilon _k(\delta )$,

$$\begin{aligned} |u_{j,\varepsilon }(x)|\le C_1\exp \left( -\frac{C_2\hbox {dist}(x,\mathcal {A}^\delta )}{\varepsilon }\right) , \ 1\le j\le k. \end{aligned}$$

For any positive integer k, fixed $0<\varepsilon <\varepsilon _k$. It is easy to see that $u_{j,\varepsilon }$ obtained in Theorem 1.1 depends on b. In the sequel, denote $u_{j,\varepsilon }$ by $u_{j,\varepsilon }^b$ to emphasize this dependence. We then shall give the asymptotic behavior of $u_{j,\varepsilon }^b$ as $b\searrow 0$, which reflects some relationship between $b>0$ and $b=0$ about Eq. (1.1). We have the following theorem about this point.

Theorem 1.2

Suppose that $(V_1)$ and $(V_2)$ are satisfied. Then for any positive integer k and for any sequence $\{b_n\}$ with $b_n\searrow 0$ as $n\rightarrow \infty$, there exists $\varepsilon _k>0$ independent of $b_n$, such that for each $0<\varepsilon <\varepsilon _k$, Eq. (1.1) possesses at least k pairs of sign-changing solutions $\pm u_{j,\varepsilon }^{b_n}$ satisfying $u_{j,\varepsilon }^{b_n}\rightharpoonup u_{j,\varepsilon }^0$ in $H^1(\mathbb {R}^3)$ as $n\rightarrow \infty$, where $j=1,2,\ldots ,k$ and $u_{j,\varepsilon }^0$ is a weak solution of the problem

$$\begin{aligned} -\varepsilon ^2a\Delta u+V(x)u=|u|^{p-2}u, \ x \in \mathbb {R}^3. \end{aligned}$$

To investigate (1.1), we will focus on the following equivalent equation by making the change of variable $x\mapsto \varepsilon x$,

$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(\varepsilon x)u=|u|^{p-2}u, \ x \in \mathbb {R}^3, \end{aligned}$$

(1.5)

whose Euler–Lagrange energy functional is

$$\begin{aligned} I_\varepsilon (u)= & {} \frac{a}{2}\int _{\mathbb {R}^3}|\nabla u|^2dx+\frac{b}{4}\left( \int _{\mathbb {R}^3}|\nabla u|^2dx\right) ^2\nonumber \\&+\frac{1}{2}\int _{\mathbb {R}^3}V(\varepsilon x)u^2dx -\frac{1}{p}\int _{\mathbb {R}^3}|u|^pdx. \end{aligned}$$

(1.6)

Set

$$\begin{aligned} \Vert u\Vert _\varepsilon =\left( \int _{\mathbb {R}^3}[a|\nabla u|^2+V(\varepsilon x)u^2]dx\right) ^{\frac{1}{2}}. \end{aligned}$$

By $(V_1)$, the norms $\Vert u\Vert _\varepsilon$ and

$$\begin{aligned} \Vert u\Vert :=\left( \int _{\mathbb {R}^3}[a|\nabla u|^2+u^2]dx\right) ^{\frac{1}{2}} \end{aligned}$$

are equivalent norms and $H^1(\mathbb {R}^3)\hookrightarrow L^t(\mathbb {R}^3)$ is continuous for each $2\le t \le 6$ and locally compact for each $2\le t <6$. Clearly, $I_\varepsilon$ is well defined on $H^1(\mathbb {R}^3)$ and $I_\varepsilon \in C^1(H^1(\mathbb {R}^3), \mathbb {R})$.

Our argument is based on variational method, which can be outlined as follows. The solutions are obtained as critical points of the energy functional (1.6) associated to Eq. (1.5). On the one hand, we remark that the non-degeneracy of sign-changing solutions are far more unclear given the progress made recently. Here we do not use non-degeneracy conditions of solutions of limiting equation, so it is not applicable to use the Lyapunov–Schmidt reduction type method or gluing method. On the other hand, since we work in the whole space, the main difficulty when dealing with this problem is the lack of compactness of Sobolev embedding. It is natural to ask how to show $I_\varepsilon$ satisfies the $(PS)_c$ condition? Generally speaking, using Rabinowitz [39], one can prove that $I_\varepsilon$ satisfies the $(PS)_c$ condition if c is smaller than the mountain pass value of the limiting functional

$$\begin{aligned} I_{\infty }(u):= & {} \frac{a}{2}\int _{\mathbb {R}^3}|\nabla u|^2dx+\frac{b}{4}\left( \int _{\mathbb {R}^3}|\nabla u|^2dx\right) ^2 +\frac{1}{2}V_{\infty }\int _{\mathbb {R}^3}u^2dx -\frac{1}{p}\int _{\mathbb {R}^3}|u|^pdx, \end{aligned}$$

where $V_{\infty }:=\liminf _{|x|\rightarrow \infty }V(x)$. Unfortunately, we have no global information on the potential V, so the argument of Rabinowitz [39] cannot be directly used here, and some new methods and techniques need to be introduced in this paper.

More precisely, to prove the main results, some arguments are in order. First, we use the penalization approach developed by Byeon and Wang [6]. In fact, following [6] we consider the penalized functional $J_\varepsilon (u)=I_\varepsilon (u)+Q_\varepsilon (u)$ with $Q_\varepsilon$ defined in the next section. In such a way, the penalized functional $J_\varepsilon$ has the advantage that it has a higher threshold for (PS) condition to hold. Then, taking advantage of an abstract minimax theorem about sign-changing solutions (see [9]) we can find that the penalized functional $J_\varepsilon$ possesses multiple sign-changing critical points. A last and critical step in such an approach is to prove that these critical points are in fact critical points of $I_\varepsilon$ for $\varepsilon$ small. To do this, we use the Moser’s iteration technique to establish a delicate $L^{\infty }$-estimation for the critical point of $J_\varepsilon$, and moreover show that the critical point of $J_\varepsilon$ is indeed a solution to the original problem. Finally, using the local Pohozaev identity we show that the concentration points of these solutions lie in $\mathcal {A}$. Let us point out that, although some similar ideas were used before for the Schrödinger equation (see [9] and [34]), the adaptation to the procedure to our problem is not trivial at all. Since the nonlocal term $(\int _{\mathbb {R}^3}|\nabla u|^2dx)\Delta u$ is involved in the equation, which brings some new difficulties such that Eq. (1.1) no longer enjoys the same good properties as the Schrödinger equation. So we need to give more finer asymptotic analysis and more detailed estimates in the present paper.

2 Multiple Sign-Changing Critical Points of Penalized Functional

Since Byeon and Wang’s penalization approach [6] guarantees the penalized functional has a higher threshold for Palais–Smale condition to hold (see Lemma 2.1). Following [6], let $\zeta \in C^{\infty }(\mathbb {R})$ be such that $0 \le \zeta (t)\le 1$ and $\zeta ^\prime (t) \ge 0$ for any $t \in \mathbb {R}$; $\zeta (t)=0$ for $t\le 0$, $\zeta (t)>0$ for $t>0$, and $\zeta (t)=1$ for $t\ge 1$. Define

$$\begin{aligned} \begin{array}{ll} \chi _\varepsilon (x)= \left\{ \begin{array}{ll} 0, &{} x \in \mathcal {D}_\varepsilon ,\\ \varepsilon ^{-6}\zeta (\hbox {dist}(x,\mathcal {D}_\varepsilon )), &{} x \not \in \mathcal {D}_\varepsilon . \end{array}\right. \end{array} \end{aligned}$$

Clearly, $\chi _\varepsilon$ is a $C^1$ function for $\varepsilon >0$ small and $\chi _\varepsilon (x)=0$ if $x \in \mathcal {D}_\varepsilon$ and $\chi _\varepsilon (x)=\varepsilon ^{-6}$ if $x \not \in (\mathcal {D}_\varepsilon )^1$. For $u \in H^1(\mathbb {R}^3)$, we introduce the penalization term

$$\begin{aligned} Q_\varepsilon (u)=\frac{1}{2\beta }\left( \int _{\mathbb {R}^3} \chi _\varepsilon (x)u^2dx-1\right) _+^\beta , \end{aligned}$$

where $2<2\beta <p$ and $(t)_+:=\max \{t,0\}$.

Set

$$\begin{aligned} J_\varepsilon (u)= & {} I_\varepsilon (u)+Q_\varepsilon (u)\\= & {} \frac{a}{2}\int _{\mathbb {R}^3}|\nabla u|^2dx+\frac{b}{4}(\int _{\mathbb {R}^3}|\nabla u|^2dx)^2 +\frac{1}{2}\int _{\mathbb {R}^3}V(\varepsilon x)u^2dx -\frac{1}{p}\int _{\mathbb {R}^3}|u|^pdx\\&+\frac{1}{2\beta }\left( \int _{\mathbb {R}^3}\chi _\varepsilon (x)u^2dx-1\right) _+^\beta , \ \forall u \in H^1(\mathbb {R}^3). \end{aligned}$$

Then, for any $\varphi \in H^1(\mathbb {R}^3)$,

$$\begin{aligned} \langle J_\varepsilon ^\prime (u),\varphi \rangle= & {} a\int _{\mathbb {R}^3}\nabla u\cdot \nabla \varphi dx+b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}\nabla u\cdot \nabla \varphi dx\\&+\int _{\mathbb {R}^3}V(\varepsilon x)u\varphi dx-\int _{\mathbb {R}^3}|u|^{p-2}u \varphi dx\\&+\left( \int _{\mathbb {R}^3}\chi _\varepsilon (x)u^2dx-1\right) _+^{\beta -1} \int _{\mathbb {R}^3}\chi _\varepsilon (x)u \varphi dx, \end{aligned}$$

whose critical point u is a solution of

$$\begin{aligned} -(a+b\int _{\mathbb {R}^3}|\nabla u|^2dx)\Delta u+V(\varepsilon x)u+\left( \int _{\mathbb {R}^3}\chi _\varepsilon (x)u^2dx-1\right) _+^{\beta -1} \chi _\varepsilon (x)u=|u|^{p-2}u.\nonumber \\ \end{aligned}$$

(2.1)

It is easy to see that if u is a critical point of $J_\varepsilon$ satisfying $Q_\varepsilon (u)=0$, then u is a solution of (1.5).

Lemma 2.1

For any $L>0$, there is $\varepsilon _L>0$ such that, for any $0<\varepsilon <\varepsilon _L$, then $J_\varepsilon$ satisfies $(PS)_c$ condition for $c<L$.

Proof

Let $\{u_n\}\subset H^1(\mathbb {R}^3)$ be such that $J_\varepsilon (u_n)\rightarrow c$ and $J_\varepsilon ^\prime (u_n)\rightarrow 0$ in $H^{-1}(\mathbb {R}^3)$. Then,

$$\begin{aligned}&o(1)+o(1)\Vert u_n\Vert +L\ge o(1)+o(1)\Vert u_n\Vert +c=J_\varepsilon (u_n)-\frac{1}{p}\langle J_\varepsilon ^\prime (u_n),u_n \rangle \\&\quad =\left( \frac{1}{2}-\frac{1}{p}\right) a\int _{\mathbb {R}^3}|\nabla u_n|^2dx+\left( \frac{1}{4}-\frac{1}{p}\right) b(\int _{\mathbb {R}^3}|\nabla u_n|^2dx)^2+\left( \frac{1}{2}-\frac{1}{p}\right) \int _{\mathbb {R}^3}V(\varepsilon x)u_n^2dx\\&\quad \quad +\frac{1}{2\beta }\left( \int _{\mathbb {R}^3}\chi _\varepsilon (x)u_n^2dx-1\right) _+^{\beta } -\frac{1}{p}\left( \int _{\mathbb {R}^3}\chi _\varepsilon (x)u_n^2dx-1\right) _+^{\beta -1} \int _{\mathbb {R}^3}\chi _\varepsilon (x)u_n^2dx, \end{aligned}$$

which implies that there exists $C_L>0$ independent of $\varepsilon$ such that $\Vert u_n\Vert \le C_L$ and $Q_\varepsilon (u_n)\le C_L$. Up to a subsequence, there exist $u \in H^1(\mathbb {R}^3)$ and $\lambda \in \mathbb {R}\backslash \mathbb {R}^-$ such that $u_n \rightharpoonup u$ in $H^1(\mathbb {R}^3)$ and $\lambda _n:=(\int _{\mathbb {R}^3}\chi _\varepsilon (x)u_n^2dx-1)_+^{\beta -1} \rightarrow \lambda$ as $n\rightarrow \infty$. By using of a standard argument, we can deduce that u is a solution of

$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(\varepsilon x)u+\lambda \chi _\varepsilon (x)u=|u|^{p-2}u. \end{aligned}$$

(2.2)

Consequently, for any $\varphi \in H^1(\mathbb {R}^3)$,

$$\begin{aligned} o(1)\Vert \varphi \Vert&=\langle J_\varepsilon ^\prime (u_n),\varphi \rangle \nonumber \\&=a\int _{\mathbb {R}^3}\nabla u_n\cdot \nabla \varphi dx+b\int _{\mathbb {R}^3}|\nabla u_n|^2dx\int _{\mathbb {R}^3}\nabla u_n\cdot \nabla \varphi dx+\int _{\mathbb {R}^3}V(\varepsilon x)u_n\varphi dx\nonumber \\&\quad -\int _{\mathbb {R}^3}|u_n|^{p-2}u_n \varphi dx+\lambda _n\int _{\mathbb {R}^3}\chi _\varepsilon (x)u_n \varphi dx-a\int _{\mathbb {R}^3}\nabla u\cdot \nabla \varphi dx\nonumber \\&\quad -b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}\nabla u\cdot \nabla \varphi dx\nonumber \\&\quad -\int _{\mathbb {R}^3}V(\varepsilon x)u\varphi dx +\int _{\mathbb {R}^3}|u|^{p-2}u \varphi dx-\lambda \int _{\mathbb {R}^3}\chi _\varepsilon (x)u \varphi dx\nonumber \\&=a\int _{\mathbb {R}^3}\nabla (u_n-u)\cdot \nabla \varphi dx+b\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx\int _{\mathbb {R}^3}\nabla u_n\cdot \nabla \varphi dx\nonumber \\&\quad +b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}\nabla (u_n-u)\cdot \nabla \varphi dx+\int _{\mathbb {R}^3}V(\varepsilon x)(u_n-u)\varphi dx\nonumber \\&\quad +\lambda \int _{\mathbb {R}^3}\chi _\varepsilon (x)(u_n-u) \varphi dx+(\lambda _n-\lambda )\int _{\mathbb {R}^3}\chi _\varepsilon (x)u_n \varphi dx\nonumber \\&\quad -\int _{\mathbb {R}^3}(|u_n|^{p-2}u_n-|u|^{p-2}u) \varphi dx. \end{aligned}$$

(2.3)

Let $r_0>0$ be such that $\mathcal {D}\subset B_{r_0}(0)$ and $\phi _\varepsilon$ be a $C^{\infty }$ cut-off function such that $0\le \phi _\varepsilon \le 1$, $\phi _\varepsilon (x)=0$ for $|x|\le \varepsilon ^{-1}r_0+1$ and $\phi _\varepsilon (x)=1$ for $|x|\ge \varepsilon ^{-1}r_0+2$, and $|\nabla \phi _\varepsilon |\le 4$ in $\mathbb {R}^3$. Taking $\varphi =\phi _\varepsilon ^2(u_n-u)$ as a test function of (2.3), by the differential mean value theorem

$$\begin{aligned} o(1)= & {} a\int _{\mathbb {R}^3}\nabla (u_n-u)\cdot \nabla [\phi _\varepsilon ^2(u_n-u)]dx+\int _{\mathbb {R}^3}V(\varepsilon x)\phi _\varepsilon ^2(u_n-u)^2 dx\nonumber \\&+\lambda \int _{\mathbb {R}^3}\chi _\varepsilon (x)\phi _\varepsilon ^2(u_n-u)^2 dx +(\lambda _n-\lambda )\int _{\mathbb {R}^3}\chi _\varepsilon (x)\phi _\varepsilon ^2u_n (u_n-u)dx\nonumber \\&-(p-1)\int _{\mathbb {R}^3}|\theta u_n+(1-\theta )u|^{p-2}\phi _\varepsilon ^2(u_n-u)^2dx\nonumber \\&+b\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\nabla u_n\cdot \nabla [\phi _\varepsilon ^2(u_n-u)]dx\nonumber \\&+b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}\nabla (u_n-u)\cdot \nabla [\phi _\varepsilon ^2(u_n-u)]dx\nonumber \\= & {} a\int _{\mathbb {R}^3}|\nabla [\phi _\varepsilon (u_n-u)]|^2 dx+\int _{\mathbb {R}^3}V(\varepsilon x)\phi _\varepsilon ^2(u_n-u)^2 dx\nonumber \\&+\lambda \int _{\mathbb {R}^3}\chi _\varepsilon (x)\phi _\varepsilon ^2(u_n-u)^2 dx\nonumber \\&+(\lambda _n-\lambda )\int _{\mathbb {R}^3}\chi _\varepsilon (x)\phi _\varepsilon ^2u_n (u_n-u)dx-(p-1)\int _{\mathbb {R}^3}|\theta u_n\nonumber \\&+(1-\theta )u|^{p-2}\phi _\varepsilon ^2(u_n-u)^2dx -a\int _{\mathbb {R}^3}(u_n-u)^2|\nabla \phi _\varepsilon |^2dx\nonumber \\&+b\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx\int _{\mathbb {R}^3}\nabla u_n\cdot \nabla [\phi _\varepsilon ^2(u_n-u)]dx\nonumber \\&+b\int _{\mathbb {R}^3}|\nabla u|^2dx \int _{\mathbb {R}^3}|\nabla [\phi _\varepsilon (u_n-u)]|^2 dx\nonumber \\&-b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}(u_n-u)^2|\nabla \phi _\varepsilon |^2dx. \end{aligned}$$

(2.4)

Since $\lambda _n \rightarrow \lambda$ as $n\rightarrow \infty$ and $|\nabla \phi _\varepsilon |^2$ possesses a compact support,

$$\begin{aligned} (\lambda _n-\lambda )\int _{\mathbb {R}^3}\chi _\varepsilon (x)\phi _\varepsilon ^2u_n (u_n-u)dx \rightarrow 0 \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {R}^3}(u_n-u)^2|\nabla \phi _\varepsilon |^2dx \rightarrow 0 \end{aligned}$$

as $n \rightarrow \infty$. Therefore, by (2.4) and $(V_1)$ we deduce that

$$\begin{aligned}&\min \{1,\alpha _1\}\Vert \phi _\varepsilon (u_n-u)\Vert ^2\le a\int _{\mathbb {R}^3}|\nabla [\phi _\varepsilon (u_n-u)]|^2 dx\nonumber \\&\quad +\int _{\mathbb {R}^3}V(\varepsilon x)\phi _\varepsilon ^2(u_n-u)^2 dx\nonumber \\&=o(1)-\lambda \int _{\mathbb {R}^3}\chi _\varepsilon (x)\phi _\varepsilon ^2(u_n-u)^2 dx\nonumber \\&\quad -b\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\nabla u_n\cdot \nabla [\phi _\varepsilon ^2(u_n-u)]dx\nonumber \\&\quad -b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}|\nabla [\phi _\varepsilon (u_n-u)]|^2 dx\nonumber \\&\quad +(p-1)\int _{\mathbb {R}^3}|\theta u_n+(1-\theta )u|^{p-2}\phi _\varepsilon ^2(u_n-u)^2dx\nonumber \\&\le o(1)-b\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\nabla u_n\cdot \nabla [\phi _\varepsilon ^2(u_n-u)]dx\nonumber \\&\quad +(p-1)\int _{\mathbb {R}^3}|\theta u_n+(1-\theta )u|^{p-2}\phi _\varepsilon ^2(u_n-u)^2dx. \end{aligned}$$

(2.5)

Note that

$$\begin{aligned}&\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\nabla u_n\cdot \nabla [\phi _\varepsilon ^2(u_n-u)] dx\\&=\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\nabla u_n\cdot [2(u_n-u)\phi _\varepsilon \nabla \phi _\varepsilon +\phi _\varepsilon ^2 \nabla (u_n-u)]dx\\&=2\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\phi _\varepsilon (u_n-u)\nabla u_n\cdot \nabla \phi _\varepsilon dx\\&\quad +\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\phi _\varepsilon ^2\nabla u_n\cdot \nabla (u_n-u)dx\\&=o(1)+\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\phi _\varepsilon ^2\nabla u \cdot \nabla (u_n-u)dx\\&\quad +\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\phi _\varepsilon ^2|\nabla (u_n-u)|^2dx\\&=o(1)+\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\phi _\varepsilon ^2|\nabla (u_n-u)|^2dx. \end{aligned}$$

Again notice that

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\int _{\mathbb {R}^3}|\nabla u_n|^2dx \ge 2\liminf \limits _{n\rightarrow \infty }\int _{\mathbb {R}^3}\nabla u_n\cdot \nabla udx-\int _{\mathbb {R}^3}|\nabla u|^2dx=\int _{\mathbb {R}^3}|\nabla u|^2dx. \end{aligned}$$

As a result, by (2.5) we see that

$$\begin{aligned}&\min \{1,\alpha _1\}\Vert \phi _\varepsilon (u_n-u)\Vert ^2\\&\quad \le o(1)-b\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\phi _\varepsilon ^2|\nabla (u_n-u)|^2dx\\&\quad \quad +(p-1)\int _{\mathbb {R}^3}|\theta u_n+(1-\theta )u|^{p-2}\phi _\varepsilon ^2(u_n-u)^2dx, \end{aligned}$$

which yields that by Hölder inequality

$$\begin{aligned}&\min \{1,\alpha _1\}\limsup \limits _{n \rightarrow \infty }\Vert \phi _\varepsilon (u_n-u)\Vert ^2\nonumber \\&\le -b\liminf \limits _{n \rightarrow \infty }\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx\cdot \liminf \limits _{n \rightarrow \infty }\int _{\mathbb {R}^3}\phi _\varepsilon ^2|\nabla (u_n-u)|^2dx\nonumber \\&\quad +(p-1)\limsup \limits _{n \rightarrow \infty }\int _{\mathbb {R}^3}|\theta u_n+(1-\theta )u|^{p-2}\phi _\varepsilon ^2(u_n-u)^2dx\nonumber \\&\le (p-1)\limsup \limits _{n \rightarrow \infty }\int _{\mathbb {R}^3}|\theta u_n+(1-\theta )u|^{p-2}\phi _\varepsilon ^2(u_n-u)^2dx\nonumber \\&\le (p-1)\limsup \limits _{n \rightarrow \infty }\Big [(\int _{|x|\ge \varepsilon ^{-1}r_0+1}|\theta u_n+(1-\theta )u|^pdx)^{\frac{p-2}{p}}\cdot (\int _{\mathbb {R}^3} \phi _\varepsilon ^p|u_n-u|^pdx)^{\frac{2}{p}}\Big ]\nonumber \\&\le C\limsup \limits _{n \rightarrow \infty }\Bigg [\Big [(\int _{|x|\ge \varepsilon ^{-1}r_0+1}|u_n|^pdx)^{\frac{p-2}{p}}+(\int _{|x|\ge \varepsilon ^{-1}r_0+1}|u|^pdx)^{\frac{p-2}{p}}\Big ]\Vert \phi _\varepsilon (u_n-u)\Vert ^2\Bigg ],\nonumber \\ \end{aligned}$$

(2.6)

where C is a positive constant which is independent of $\varepsilon$ and n. Since $Q_\varepsilon (u_n)\le C_L$ and $(\mathcal {D}_\varepsilon )^1 \subset B_{\varepsilon ^{-1}r_0+1}(0)$,

$$\begin{aligned} \int _{|x|\ge \varepsilon ^{-1}r_0+1}u_n^2dx\le [1+(2\beta C_L)^{\frac{1}{\beta }}]\varepsilon ^6. \end{aligned}$$

Moreover, by Fatou lemma,

$$\begin{aligned} \int _{|x|\ge \varepsilon ^{-1}r_0+1}u^2dx\le [1+(2\beta C_L)^{\frac{1}{\beta }}]\varepsilon ^6. \end{aligned}$$

Since $4<p<6$, by the interpolation inequality, there exists $t \in (0,1)$ with $\frac{1}{p}=\frac{t}{2}+\frac{1-t}{6}$ such that

$$\begin{aligned} \Vert u\Vert _p^p \le \Vert u\Vert _2^{tp}\Vert u\Vert _6^{(1-t)p}\le C\Vert u\Vert _2^{tp}\Vert u\Vert ^{(1-t)p}. \end{aligned}$$

Thus,

$$\begin{aligned}&\int _{|x|\ge \varepsilon ^{-1}r_0+1}|u_n|^pdx\le C(\int _{|x|\ge \varepsilon ^{-1}r_0+1}|u_n|^2dx)^{\frac{tp}{2}}\cdot \Vert u_n\Vert ^{(1-t)p}\\&\quad \le C\Big ([1+(2\beta C_L)^{\frac{1}{\beta }}]\varepsilon ^6 \Big )^{\frac{tp}{2}}C_L^{(1-t)p}:= \widetilde{C}_L\varepsilon ^{3tp}. \end{aligned}$$

Similarly,

$$\begin{aligned} \int _{|x|\ge \varepsilon ^{-1}r_0+1}|u|^pdx\le \widetilde{C}_L\varepsilon ^{3tp}, \end{aligned}$$

where $\widetilde{C}_L>0$ independent of $\varepsilon$. Since $2C(p-1)\cdot \Big (\widetilde{C}_L\varepsilon ^{3tp}\Big )^{\frac{p-2}{p}}\rightarrow 0$ as $\varepsilon \rightarrow 0$, so there exists $\varepsilon _L>0$ such that $0<\varepsilon <\varepsilon _L$,

$$\begin{aligned} 2C(p-1)\cdot \Big (\widetilde{C}_L\varepsilon ^{3tp}\Big )^{\frac{p-2}{p}}<\frac{1}{2}\min \{1,\alpha _1\}. \end{aligned}$$

Then by (2.6) one has $\limsup \limits _{n\rightarrow \infty }\Vert \phi _\varepsilon (u_n-u)\Vert =0$. Similarly, if we choose $\varphi =(1-\phi _\varepsilon )^2(u_n-u)$ as a test function of (2.3), we can conclude that $\limsup \limits _{n\rightarrow \infty }\Vert (1-\phi _\varepsilon )(u_n-u)\Vert =0$. Hence, $u_n \rightarrow u$ in $H^1(\mathbb {R}^3)$. This completes the proof. $\square$

In the following, we will use the abstract minimax Theorem 3.2 in [9] to obtain multiple sign-changing critical points of the penalized functional $J_\varepsilon$. To this end, set

$$\begin{aligned} P_+:=\{u \in H^1(\mathbb {R}^3): u\ge 0\} \end{aligned}$$

and

$$\begin{aligned} P_-:=\{u \in H^1(\mathbb {R}^3) : u\le 0\}. \end{aligned}$$

For $\sigma >0$, let

$$\begin{aligned} P_+^\sigma :=\{u \in H^1(\mathbb {R}^3): \hbox {dist}_{H^1}(u,P_+)<\sigma \} \\ \hbox {and} \ \ P_-^\sigma :=\{u \in H^1(\mathbb {R}^3): \hbox {dist}_{H^1}(u,P_-)<\sigma \}, \end{aligned}$$

where $\hbox {dist}_{H^1}(u,A):=\inf _{v \in A}\Vert u-v\Vert$ for $u \in H^1(\mathbb {R}^3)$ and $A \subset H^1(\mathbb {R}^3)$. Clearly, $P_-^\sigma =-P_+^\sigma$. If we take $X=H^1(\mathbb {R}^3)$, $P=P_+^\sigma$, $J=J_\varepsilon$ and $W=P_+^\sigma \cup P_-^\sigma$ in Theorem 3.2 in [9], it is easy to see that W is an open and symmetric subset of $H^1(\mathbb {R}^3)$ and $H^1(\mathbb {R}^3)\setminus W$ contains only sign-changing functions. Furthermore, from the fact that 0 is a strict local minimum point of $J_\varepsilon$ we know that the constant $c^*$ in Theorem 3.2 in [9] satisfies $c^*=\inf \limits _{\partial (P_-^\sigma )\cap \partial (P_+^\sigma )}J_\varepsilon >0$ for small $\sigma >0$. Since $0 \in \mathcal {A}$, we have $B_1(0) \subset \mathcal {D}_\varepsilon$ for small $\varepsilon >0$. Let $u \in H_0^1(B_1(0))$, define

$$\begin{aligned} \Gamma _0(u)= & {} \frac{a}{2}\int _{B_1(0)}|\nabla u|^2dx+\frac{1}{2}\alpha _2\int _{B_1(0)}u^2dx\\&+\frac{b}{4}\left( \int _{\mathbb {R}^3}|\nabla \tilde{u}|^2dx\right) ^2-\frac{1}{p}\int _{B_1(0)}|u|^pdx, \end{aligned}$$

where $\tilde{u}=u$ on $B_1(0)$ and $\tilde{u}\equiv 0$ on $B_1^c(0)$. Denote an orthonormal basis of $H_0^1(B_1(0))$ by $\{e_n\}$ and define $E_n:=span\{e_1,\ldots ,e_n\}$. Since $p>4$, we derive that there exists an increasing sequence of positive numbers $\{R_n\}$ such that

$$\begin{aligned} \Gamma _0(u)<0, \ \forall u \in E_n, \ \Vert u\Vert \ge R_n. \end{aligned}$$

Define $\varphi _n\in C(B_n,H_0^1(B_1(0)))$ by

$$\begin{aligned} \varphi _n(t):=R_n\sum \limits _{j=1}^nt_je_j, \ t=(t_1,\ldots ,t_n)\in B_n, \end{aligned}$$

where $B_n=\{x \in \mathbb {R}^3:|x|\le 1\}$ comes from Theorem 3.2 in [9]. Then it is easy to prove that $\varphi _n$ satisfies (1)–(3) in Theorem 3.2 in [9]. The genus of a closed symmetric subset A is denoted by $\gamma (A)$. For $j\in \mathbb {N}$, define

$$\begin{aligned} c_j^\varepsilon =\inf _{B\in \Lambda _j}\sup _{u\in B\setminus W} J_\varepsilon (u) \, \hbox {and} \, \widetilde{c}_j=\inf _{B\in \widetilde{\Lambda }_j}\sup _{u\in B\setminus W} \Gamma _0(u), \end{aligned}$$

where

$$\begin{aligned} \Lambda _j:=\{B|B=\varphi (B_n\setminus Y) \ \hbox {for} \ \varphi \in G_n, n\ge j, \ \hbox {and open} \ Y\subset B_n\\ \hbox { such that} \ -Y=Y \ \hbox {and} \ \gamma (\overline{Y})\le n-j\} \end{aligned}$$

and

$$\begin{aligned} \widetilde{\Lambda }_j:=\{B|B=\varphi (B_n\setminus Y) \ \hbox {for} \ \varphi \in \widetilde{G}_n, n\ge j, \ \hbox {and open} \ Y\subset B_n\\ \hbox { such that} \ -Y=Y \ \hbox {and} \ \gamma (\overline{Y})\le n-j\} \end{aligned}$$

and

$$\begin{aligned} G_n:=\{\varphi |\varphi \in C(B_n, H^1(\mathbb {R}^3)), \varphi (-t)=-\varphi (t) \ \hbox {for any} \ t\in B_n, \varphi |_{\partial B_n}=\varphi _n|_{\partial B_n}\} \end{aligned}$$

and

$$\begin{aligned} \widetilde{G}_n:=\{\varphi |\varphi \in C(B_n, H^1_0(B_1(0))), \varphi (-t)=-\varphi (t) \ \hbox {for any} \ t\in B_n, \varphi |_{\partial B_n}=\varphi _n|_{\partial B_n}\}. \end{aligned}$$

Then,

$$\begin{aligned} 0<c_2^\varepsilon \le c_3^\varepsilon \le \cdots , \ \text{ and } \ \tilde{c}_2\le \tilde{c}_3\le \cdots . \end{aligned}$$

(2.7)

Taking into account of $B_1(0)\subset \mathcal {D}_\varepsilon$ and $(V_1)$, when $\varepsilon >0$ sufficiently small, we deduce that $J_\varepsilon (u)\le \Gamma _0(u)$ for any $u \in H_0^1(B_1(0))$, which together with $\widetilde{\Lambda }_j\subset \Lambda _j$ we have

$$\begin{aligned} 0<c_j^\varepsilon \le \tilde{c}_j, \ \forall j\ge 2, \end{aligned}$$

(2.8)

for sufficiently small $\varepsilon >0$. Arguing as in the proof of Theorem 3.3 in [9, 10], we have the following lemma.

Lemma 2.2

There exists $\sigma _0>0$ such that for any $0<\sigma <\sigma _0$ and any $L>0$, when $0<\varepsilon <\varepsilon _L$, $P_+^\sigma$ is an admissible invariant set with respect to $J_\varepsilon$ for $c<L$.

Proof

Let an operator A on $H^1(\mathbb {R}^3)$ be such that for any $u \in H^1(\mathbb {R}^3)$, $w=A(u)$ is the unique solution of the following equation

$$\begin{aligned}&-(a+b\int _{\mathbb {R}^3}|\nabla u|^2dx)\Delta w+V(\varepsilon x)w\nonumber \\&\qquad +\left( \int _{\mathbb {R}^3}\chi _\varepsilon (x)u^2dx-1\right) _+^{\beta -1}\chi _\varepsilon w=|u|^{p-2}u. \end{aligned}$$

(2.9)

Clearly, A is odd on $H^1(\mathbb {R}^3)$. In the sequel, we divide the proof into several steps.

Step 1: A is well defined and continuous on $H^1(\mathbb {R}^3)$. Indeed, set

$$\begin{aligned} \xi (u)=\left( \int _{\mathbb {R}^3}\chi _\varepsilon (x)u^2dx-1\right) _+^{\beta -1}\ge 0. \end{aligned}$$

It is easy to see that A is well defined on $H^1(\mathbb {R}^3)$. Let $u_n\rightarrow u$ in $H^1(\mathbb {R}^3)$, then

$$\begin{aligned}&\min \{1,\alpha _1\}\Vert A(u_n)-A(u)\Vert ^2\nonumber \\&\le a\int _{\mathbb {R}^3}|\nabla (A(u_n)-A(u))|^2dx +\int _{\mathbb {R}^3}V(\varepsilon x)|A(u_n)-A(u)|^2dx\nonumber \\&=\int _{\mathbb {R}^3}(|u_n|^{p-2}u_n-|u|^{p-2}u)(A(u_n)-A(u))dx\nonumber \\&\quad -b\int _{\mathbb {R}^3}|\nabla u_n|^2dx \int _{\mathbb {R}^3}\nabla (A(u_n))\cdot \nabla (A(u_n)-A(u))dx\nonumber \\&\quad -\xi (u_n)\int _{\mathbb {R}^3}\chi _\varepsilon (x)A(u_n)\cdot (A(u_n)-A(u))dx\nonumber \\&\quad +b\int _{\mathbb {R}^3}|\nabla u|^2dx \int _{\mathbb {R}^3}\nabla (A(u))\cdot \nabla (A(u_n)-A(u))dx\nonumber \\&\quad +\xi (u)\int _{\mathbb {R}^3}\chi _\varepsilon (x)A(u)\cdot (A(u_n)-A(u))dx\nonumber \\&=\int _{\mathbb {R}^3}(|u_n|^{p-2}u_n-|u|^{p-2}u)(A(u_n)-A(u))dx\nonumber \\&\quad -\xi (u_n) \int _{\mathbb {R}^3}\chi _\varepsilon (x)|A(u_n)-A(u)|^2dx\nonumber \\&\quad -(\xi (u_n)-\xi (u))\int _{\mathbb {R}^3}\chi _\varepsilon (x)A(u)\cdot (A(u_n)-A(u))dx\nonumber \\&\quad -b\int _{\mathbb {R}^3}|\nabla u_n|^2dx \int _{\mathbb {R}^3}|\nabla (A(u_n)-A(u))|^2dx\nonumber \\&\quad -b\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx \int _{\mathbb {R}^3}\nabla (A(u))\cdot \nabla (A(u_n)-A(u))dx\nonumber \\&\le \int _{\mathbb {R}^3}\big ||u_n|^{p-2}u_n-|u|^{p-2}u \big |\big |A(u_n)-A(u)\big |dx\nonumber \\&\quad +|\xi (u_n)-\xi (u)|\int _{\mathbb {R}^3}\chi _\varepsilon (x)|A(u)|\cdot |A(u_n)-A(u)|dx\nonumber \\&\quad +b\big |\int _{\mathbb {R}^3}(|\nabla u_n|^2-|\nabla u|^2)dx\big | \cdot \int _{\mathbb {R}^3}\big |\nabla (A(u))\big | \big |\nabla (A(u_n)-A(u))\big |dx.\nonumber \\ \end{aligned}$$

(2.10)

By Hölder inequality and differential mean value theorem, there exists $\theta \in (0,1)$ such that

$$\begin{aligned}&\int _{\mathbb {R}^3}\big ||u_n|^{p-2}u_n-|u|^{p-2}u \big |\big |A(u_n)-A(u)\big |dx\nonumber \\&=\int _{\mathbb {R}^3}\Big |(p-1)|u+\theta (u_n-u)|^{p-2}\cdot (u_n-u) \Big |\big |A(u_n)-A(u)\big |dx\nonumber \\&\le (p-1)\Big (\int _{\mathbb {R}^3}|u+\theta (u_n-u)|^p\Big )^{\frac{p-2}{p}}\cdot \Big (\int _{\mathbb {R}^3}|u_n-u|^p\Big )^{\frac{1}{p}}\cdot \Big (\int _{\mathbb {R}^3}|A(u_n)-A(u)|^p\Big )^{\frac{1}{p}}\nonumber \\&\le C \Vert u_n-u\Vert _p\cdot \Vert A(u_n)-A(u)\Vert . \end{aligned}$$

(2.11)

By virtue of Sobolev embedding, we can prove that

$$\begin{aligned} \xi (u_n)\rightarrow \xi (u) \ \hbox {as} \ n\rightarrow \infty . \end{aligned}$$

(2.12)

As a result, (2.10)–(2.12) yields that

$$\begin{aligned}&\min \{1,\alpha _1\}\Vert A(u_n)-A(u)\Vert ^2\\&\le C \Vert u_n-u\Vert _p\cdot \Vert A(u_n)-A(u)\Vert +|\xi (u_n)-\xi (u)|\cdot \Vert A(u_n)-A(u)\Vert \\&\qquad +\frac{b}{a}\Big |\int _{\mathbb {R}^3}|\nabla u_n|^2dx -\int _{\mathbb {R}^3}|\nabla u|^2dx\Big | \cdot \Vert A(u)\Vert \cdot \Vert A(u_n)-A(u)\Vert . \end{aligned}$$

Consequently, $\Vert A(u_n)-A(u)\Vert \rightarrow 0$ as $n \rightarrow \infty$ and so A is continuous on $H^1(\mathbb {R}^3)$.

Step 2: For any $u \in H^1(\mathbb {R}^3)$,

$$\begin{aligned} \langle J_\varepsilon ^\prime (u), u-A(u)\rangle&=a \int _{\mathbb {R}^3}|\nabla (u-A(u))|^2dx +b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}|\nabla (u-A(u))|^2dx\\&\qquad +\int _{\mathbb {R}^3}V(\varepsilon x)|u-A(u)|^2dx+\xi (u) \int _{\mathbb {R}^3}\chi _\varepsilon (x)|u-A(u)|^2dx\\&=\Vert u-A(u)\Vert _\varepsilon ^2+\xi (u)\int _{\mathbb {R}^3}\chi _\varepsilon (x)|u-A(u)|^2dx\\&\qquad +b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}|\nabla (u-A(u))|^2dx. \end{aligned}$$

Furthermore, there is a positive constant C such that

$$\begin{aligned} \Vert J_\varepsilon ^\prime (u)\Vert _{H^{-1}(\mathbb {R}^3)}\le C\Vert u-A(u)\Vert \cdot (1+\Vert u\Vert ^2+\Vert u\Vert ^{2\beta -2}), \ \forall u \in H^1(\mathbb {R}^3). \end{aligned}$$

In fact, since A(u) solves (2.9),

$$\begin{aligned}&\langle A(u),\varphi \rangle _\varepsilon +b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}\nabla (A(u))\cdot \nabla \varphi dx +\xi (u)\int _{\mathbb {R}^3}\chi _\varepsilon (x)A(u)\varphi dx\\&\quad =\int _{\mathbb {R}^3}|u|^{p-2}u\varphi dx, \ \forall \varphi \in H^1(\mathbb {R}^3), \end{aligned}$$

where

$$\begin{aligned} \langle u,\varphi \rangle _\varepsilon =\int _{\mathbb {R}^3}[a\nabla u\cdot \nabla \varphi +V(\varepsilon x)u \varphi ]dx. \end{aligned}$$

Hence,

$$\begin{aligned} \langle J_\varepsilon ^\prime (u), \varphi \rangle= & {} \langle u,\varphi \rangle _\varepsilon +b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}\nabla u\cdot \nabla \varphi dx\\&+\xi (u)\int _{\mathbb {R}^3}\chi _\varepsilon (x)u\varphi dx-\int _{\mathbb {R}^3}|u|^{p-2}u\varphi dx\\= & {} \langle u-A(u),\varphi \rangle _\varepsilon +b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}\nabla (u-A(u))\cdot \nabla \varphi dx\\&+\xi (u)\int _{\mathbb {R}^3}\chi _\varepsilon (x)(u-A(u))\varphi dx, \end{aligned}$$

which means that the inequality holds. Moreover,

$$\begin{aligned}&b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}\nabla (u-A(u))\cdot \nabla \varphi dx \le \frac{b}{a}\int _{\mathbb {R}^3}|\nabla u|^2dx \cdot \Vert u-A(u)\Vert \cdot \Vert \varphi \Vert \\&\quad \le \frac{b}{a^2}\Vert u\Vert ^2 \cdot \Vert u-A(u)\Vert \cdot \Vert \varphi \Vert \end{aligned}$$

and

$$\begin{aligned} |\xi (u)\int _{\mathbb {R}^3}\chi _\varepsilon (x)(u-A(u))\varphi dx|\le C\Vert u\Vert ^{2\beta -2}\Vert u-A(u)\Vert \Vert \varphi \Vert . \end{aligned}$$

Therefore, for any $\varphi \in H^1(\mathbb {R}^3)$,

$$\begin{aligned} \langle J_\varepsilon ^\prime (u), \varphi \rangle\le & {} C\Vert u-A(u)\Vert \cdot \Vert \varphi \Vert +\frac{b}{a^2}\Vert u\Vert ^2 \cdot \Vert u-A(u)\Vert \cdot \Vert \varphi \Vert \\&+C\Vert u\Vert ^{2\beta -2}\cdot \Vert u-A(u)\Vert \cdot \Vert \varphi \Vert \\\le & {} C\Vert u-A(u)\Vert \cdot (1+\Vert u\Vert ^2+\Vert u\Vert ^{2\beta -2})\cdot \Vert \varphi \Vert , \end{aligned}$$

which indicates the proof holds.

Step 3: There exists $\sigma _0>0$ such that for $\sigma \in (0,\sigma _0)$, $A(\partial (P_{-}^\sigma )) \subset P_{-}^\sigma$, $A(\partial (P_{+}^\sigma ))\subset P_{+}^\sigma$. We only prove that $A(\partial (P_{-}^\sigma )) \subset P_{-}^\sigma$. For any $u \in H^1(\mathbb {R}^3)$, set $w=A(u)$. By the embedding $H^1(\mathbb {R}^3) \hookrightarrow L^t(\mathbb {R}^3)$ for $2\le t \le 6$ and $(V_1)$, we have

$$\begin{aligned}&\hbox {dist}_{H^1}(w,P_{-})\Vert w^+\Vert =\inf \limits _{v \in P_{-}}\Vert w-v\Vert \cdot \Vert w^+\Vert \le \Vert w-w^{-}\Vert \cdot \Vert w^{+}\Vert = \Vert w^+\Vert ^2\\&\qquad \qquad \qquad \qquad \qquad \quad \le \max \left\{ 1, \frac{1}{\alpha _1}\right\} \Vert w^+\Vert _\varepsilon ^2=\max \left\{ 1, \frac{1}{\alpha _1}\right\} \langle w,w^+\rangle _\varepsilon \\&\qquad \qquad \qquad \qquad \qquad \quad =\max \left\{ 1, \frac{1}{\alpha _1}\right\} \Big [\int _{\mathbb {R}^3}|u|^{p-2}uw^+dx-\xi (u) \int _{\mathbb {R}^3}\chi _\varepsilon (x)ww^+dx\\&\qquad \qquad \qquad \qquad \qquad \qquad -b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}\nabla w\cdot \nabla w^+dx\Big ]\\&\qquad \qquad \qquad \qquad \qquad \quad =\max \left\{ 1, \frac{1}{\alpha _1}\right\} \Big [\int _{\mathbb {R}^3}|u|^{p-2}uw^+dx-\xi (u) \int _{\mathbb {R}^3}\chi _\varepsilon (x)|w^+|^2dx\\&\qquad \qquad \qquad \qquad \qquad \qquad -b\int _{\mathbb {R}^3}|\nabla u|^2dx\int _{\mathbb {R}^3}|\nabla w^+|^2dx\Big ]\\&\qquad \qquad \qquad \qquad \qquad \quad \le \max \left\{ 1, \frac{1}{\alpha _1}\right\} \int _{\mathbb {R}^3}|u|^{p-2}uw^+dx\\&\qquad \qquad \qquad \qquad \qquad \quad =\max \left\{ 1, \frac{1}{\alpha _1}\right\} \int _{\mathbb {R}^3}|u|^{p-2}u^+w^+dx\\&\qquad \qquad \qquad \qquad \qquad \quad \le \max \left\{ 1, \frac{1}{\alpha _1}\right\} \Vert u^+\Vert _p^{p-1}\cdot \Vert w^+\Vert _p\\&\qquad \qquad \qquad \qquad \qquad \quad \le \max \left\{ 1, \frac{1}{\alpha _1}\right\} \Big (\hbox {dist}_{L^p}(u,P_{-})\Big )^{p-1}\Vert w^+\Vert \\&\qquad \qquad \qquad \qquad \qquad \quad \le \max \left\{ 1, \frac{1}{\alpha _1}\right\} \Big (\hbox {dist}_{H^1}(u,P_{-})\Big )^{p-1}\Vert w^+\Vert _p\\&\qquad \qquad \qquad \qquad \qquad \quad \le C \Big (\hbox {dist}_{H^1}(u,P_{-})\Big )^{p-1}\Vert w^+\Vert , \end{aligned}$$

which implies that $\hbox {dist}_{H^1}(w,P_{-}) \le C \sigma ^{p-1}$. Hence, for $\sigma >0$ small, $\hbox {dist}_{H^1}(w,P_{-}) \le C \sigma ^{p-1}< \sigma$.

Since A is only continuous, we shall have a locally Lipschitz perturbation of A. Let $\mathcal {K}$ be the set of fixed points of A, i.e., the set of critical points of $J_\varepsilon$, and set $E_0:=H^1(\mathbb {R}^3)\backslash \mathcal {K}$.

Step 4: There exists a locally Lipschitz continuous operator $B: E_0 \rightarrow H^1(\mathbb {R}^3)$ such that

$\mathrm{(i)}$ $B(\partial (P_{+}^\sigma )) \subset P_{+}^\sigma$ and $B(\partial (P_{-}^\sigma )) \subset P_{-}^\sigma$ for $\sigma \in (0,\sigma _0)$.

$\mathrm{(ii)}$ $\frac{1}{2}\Vert u-B(u)\Vert \le \Vert u-A(u)\Vert \le 2\Vert u-B(u)\Vert , \ \forall u \in E_0$.

$\mathrm{(iii)}$ $\langle J_\varepsilon ^\prime (u), u-B(u)\rangle \ge \frac{1}{2}\Vert u-A(u)\Vert ^2, \ \forall u \in E_0$.

$\mathrm{(iv)}$ B is odd.

With a similar argument as the proof of Lemma 4.1 in [3] and Lemma 2.1 in [4], we can complete the proof. We omit it here.

Step 5: Let $0<\varepsilon <\varepsilon _L$ and $c<L$. Let $\mathcal {N}$ be a symmetric closed neighborhood of $K_c$. Then, there exists a positive constant $\tau _0$ such that $0<\tau<\tau ^\prime <\tau _0$, there exists a continuous map $\zeta : [0,1] \times H^1(\mathbb {R}^3) \rightarrow H^1(\mathbb {R}^3)$ satisfying

$\mathrm{(i)}$ $\zeta (0,u)=u$ for all $u \in H^1(\mathbb {R}^3)$.

$\mathrm{(ii)}$ $\zeta (t,u)=u$ for $t \in [0,1]$, $J_\varepsilon (u) \not \in [c-\tau ^\prime ,c+\tau ^\prime ]$.

$\mathrm{(iii)}$ $\zeta (t,-u)=-\zeta (t,u)$ for all $t \in [0,1]$ and $u \in H^1(\mathbb {R}^3)$.

$\mathrm{(iv)}$ $\zeta (1,(J_\varepsilon )^{c+\tau }\backslash \mathcal {N}) \subset (J_\varepsilon )^{c-\tau }$.

$\mathrm{(v)}$ $\zeta (t,\partial (P_{+}^\sigma ))\subset P_{+}^\sigma$, $\zeta (t,\partial (P_{-}^\sigma ))\subset P_{-}^\sigma$, $\zeta (t, P_{+}^\sigma )\subset P_{+}^\sigma$, $\zeta (t,P_{-}^\sigma )\subset P_{-}^\sigma$, $t \in [0,1]$.

By Lemma 2.1 and step 4, arguing as the proof of Lemma 3.5 in [32] or Lemma 3.6 in [33], we can complete the proof. We omit it here.

As a result, let D be a closed symmetric neighborhood of $K_c\backslash W$. Then $\mathcal {N}=D\cup \overline{P_{+}^\sigma }\cup \overline{P_{-}^\sigma }$ is a closed symmetric neighborhood of $K_c$. Taking $\eta (\cdot )=\zeta (1,\cdot )$ in Definition 2.3(b) in [32], we complete the proof. $\square$

By using of (2.7)–(2.8) and Theorem 3.2 in [9], the following lemma holds.

Lemma 2.3

For any $k \in \mathbb {N}^+$, there exists $\varepsilon _k^\prime >0$ such that, for $0<\varepsilon < \varepsilon _k^\prime$, $J_\varepsilon$ has at least k pairs of sign-changing critical points $\{\pm u_{j,\varepsilon } : 1\le j\le k\}$ satisfying that $J_\varepsilon (u_{j,\varepsilon })=c_{j+1}^\varepsilon \le \widetilde{c}_{k+1}$, $1\le j\le k$.

3 Proof of Theorem 1.1

In the sequel, we will prove that the sign-changing critical points $\{u_{j,\varepsilon }\}$ obtained in Lemma 2.3 are solutions of (1.5).

Lemma 3.1

For any $k \in \mathbb {N}^+$ and $0<\varepsilon < \varepsilon _k^\prime$, there exist a constant $\rho >0$ depending only on $\alpha _1$, p and a constant $\eta _k>0$ independent of $\varepsilon$ such that

$$\begin{aligned} \rho \le \Vert u_{j,\varepsilon }\Vert \le \eta _k \ \hbox {and} \ Q_\varepsilon (u_{j,\varepsilon })\le \eta _k, \ 1\le j\le k. \end{aligned}$$

Proof

By Lemma 2.3, $2<2\beta <p$ and $(V_1)$, we have

$$\begin{aligned} \tilde{c}_{k+1}\ge & {} c_{j+1}^\varepsilon =J_\varepsilon (u_{j,\varepsilon })-\frac{1}{p}\langle J_\varepsilon ^\prime (u_{j,\varepsilon }),u_{j,\varepsilon }\rangle \\= & {} \Big (\frac{1}{2}-\frac{1}{p}\Big )a\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx+\Big (\frac{1}{4}-\frac{1}{p}\Big )b \Big (\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx\Big )^2\\&+\Big (\frac{1}{2}-\frac{1}{p}\Big )\int _{\mathbb {R}^3}V(\varepsilon x)|u_{j,\varepsilon }|^2dx\\&+\frac{1}{2\beta }\Big (\int _{\mathbb {R}^3}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx-1\Big )_{+}^\beta -\frac{1}{p}\Big (\int _{\mathbb {R}^3}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx-1\Big )_{+}^{\beta -1}\nonumber \\&\cdot \int _{\mathbb {R}^3}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx\\\ge & {} \Big (\frac{1}{2}-\frac{1}{p}\Big )\min \{1,\alpha _1\}\Vert u_{j,\varepsilon }\Vert ^2 +\frac{1}{2\beta }\Big (\int _{\mathbb {R}^3}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx-1\Big )_{+}^\beta \\&-\frac{1}{p}\Big (\int _{\mathbb {R}^3}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx-1\Big )_{+}^{\beta -1}\cdot \int _{\mathbb {R}^3}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx, \end{aligned}$$

which means that there exists $\eta _k>0$ independent of $\varepsilon$ such that

$$\begin{aligned} \Vert u_{j,\varepsilon }\Vert \le \eta _k \ \hbox {and} \ Q_\varepsilon (u_{j,\varepsilon })\le \eta _k. \end{aligned}$$

Furthermore, from the fact that $\langle J_\varepsilon ^\prime (u_{j,\varepsilon }),u_{j,\varepsilon }\rangle =0$ one has

$$\begin{aligned}&\min \{1,\alpha _1\}\Vert u_{j,\varepsilon }\Vert ^2\le a\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx +\int _{\mathbb {R}^3}V(\varepsilon x)|u_{j,\varepsilon }|^2dx+b\Big (\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx\Big )^2\\&\qquad +\Big (\int _{\mathbb {R}^3}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx-1\Big )_{+}^{\beta -1}\cdot \int _{\mathbb {R}^3}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx\\&\qquad =\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^pdx\le C(p)\Vert u_{j,\varepsilon }\Vert ^p, \end{aligned}$$

which yields that

$$\begin{aligned}\Vert u_{j,\varepsilon }\Vert ^{p-2}\ge \frac{1}{C(p)}\min \{1,\alpha _1\},\end{aligned}$$

then,

$$\begin{aligned} \Vert u_{j,\varepsilon }\Vert \ge \Big (\frac{1}{C(p)}\min \{1,\alpha _1\}\Big )^{\frac{1}{p-2}}:=\rho >0, \ \forall 1\le j \le k. \end{aligned}$$

This completes the proof. $\square$

By Lemma 3.1, arguing as in the proof of Lemmas 4.2-4.3 in [9], with the aid of the definition of $\chi _\varepsilon$ and Moser’s iteration we can prove that the next two lemmas hold.

Lemma 3.2

For any $\delta >0$,

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0}\Vert u_{j,\varepsilon }\Vert _{L^{\infty }(\mathbb {R}^3\backslash (\mathcal {D}_\varepsilon )^\delta )}=0, \ 1 \le j\le k. \end{aligned}$$

Proof

From the definition of $\zeta (\cdot )$ we know that for any $\delta >0$, there exists $C_\delta >0$ such that as $t\ge \delta$, $\zeta (t)\ge C_\delta$. Consequently, by Lemma 3.1 we get that

$$\begin{aligned}&(2\beta \eta _k)^{\frac{1}{\beta }}+1\ge \int _{\mathbb {R}^3}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx \ge \int _{\mathbb {R}^3\backslash (\mathcal {D}_\varepsilon )^\delta }\chi _\varepsilon (x)|u_{j,\varepsilon }|^2dx\\&\quad =\int _{\mathbb {R}^3\backslash (\mathcal {D}_\varepsilon )^\delta } \varepsilon ^{-6}\zeta (\hbox {dist}(x,\mathcal {D}_\varepsilon ))|u_{j,\varepsilon }|^2dx\\&\quad \ge \varepsilon ^{-6}C_\delta \int _{\mathbb {R}^3\backslash (\mathcal {D}_\varepsilon )^\delta }|u_{j,\varepsilon }|^2dx, \end{aligned}$$

and so

$$\begin{aligned} \int _{\mathbb {R}^3\backslash (\mathcal {D}_\varepsilon )^\delta }|u_{j,\varepsilon }|^2dx \le \frac{(2\beta \eta _k)^{\frac{1}{\beta }}+1}{C_\delta }\cdot \varepsilon ^6:=C(k,\delta )\varepsilon ^6, \ \forall 1\le j\le k. \end{aligned}$$

(3.1)

We claim that $\Vert u_{j,\varepsilon }\Vert _{\infty }\le C$. Indeed, let $T>0$ and set $u^T(x)=u(x)$ if $|u(x)|\le T$, $u^T(x)=T$ if $u(x)\ge T$ and $u^T(x)=-T$ if $u(x)\le -T$, Take $\varphi =u_{j,\varepsilon }|u_{j,\varepsilon }^T|^{2k-2} \ (k\ge 1)$ be a test function, then

$$\begin{aligned}&a\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla \varphi dx+ b\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla \varphi dx\nonumber \\&\quad +\int _{\mathbb {R}^3}V(\varepsilon x)u_{j,\varepsilon }\varphi dx +\xi (u_{j,\varepsilon })\int _{\mathbb {R}^3}\chi _\varepsilon (x)u_{j,\varepsilon }\varphi dx\nonumber \\&\quad =\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{p-2}u_{j,\varepsilon } \varphi dx. \end{aligned}$$

(3.2)

Clearly, by Hölder inequality we have

$$\begin{aligned}&\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{p-2}u_{j,\varepsilon } \varphi dx =\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{p-2}|u_{j,\varepsilon }|^2|u_{j,\varepsilon }^T|^{2k-2}dx\nonumber \\&\quad \le \Big (\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^6\Big )^{\frac{p-2}{6}}\cdot \Big (\int _{\mathbb {R}^3}(u_{j,\varepsilon }|u_{j,\varepsilon }^T|^{k-1})^{6\cdot \frac{2}{8-p}}dx\Big )^{\frac{8-p}{6}}\nonumber \\&\quad \le C\Big (\int _{\mathbb {R}^3}(u_{j,\varepsilon }|u_{j,\varepsilon }^T|^{k-1})^{6\cdot \frac{2}{8-p}}dx\Big )^{\frac{8-p}{6}}. \end{aligned}$$

(3.3)

On the other hand, by $(V_1)$ we get

$$\begin{aligned}&a\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla \varphi dx+ b\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla \varphi dx\nonumber \\&\quad +\int _{\mathbb {R}^3}V(\varepsilon x)u_{j,\varepsilon }\varphi dx +\xi (u_{j,\varepsilon })\int _{\mathbb {R}^3}\chi _\varepsilon (x)u_{j,\varepsilon }\varphi dx\nonumber \\&\ge (a+b\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx)\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla [u_{j,\varepsilon }|u_{j,\varepsilon }^T|^{2k-2}]dx\nonumber \\&\quad +\alpha _1\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^2|u_{j,\varepsilon }^T|^{2k-2}dx +\xi (u_{j,\varepsilon })\int _{\mathbb {R}^3}\chi _\varepsilon (x) |u_{j,\varepsilon }|^2|u_{j,\varepsilon }^T|^{2k-2}dx\nonumber \\&\ge a\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla [u_{j,\varepsilon }|u_{j,\varepsilon }^T|^{2k-2}]dx\nonumber \\&=a\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot [(2k-2)|u_{j,\varepsilon }^T|^{2k-3}\frac{u_{j,\varepsilon }^T}{|u_{j,\varepsilon }^T|}u_{j,\varepsilon }\cdot \nabla u_{j,\varepsilon }^T +|u_{j,\varepsilon }^T|^{2k-2}\nabla u_{j,\varepsilon }]dx\nonumber \\&=a(2k-2)\int _{\mathbb {R}^3}|u_{j,\varepsilon }^T|^{2k-4}u_{j,\varepsilon }^Tu_{j,\varepsilon }\nabla u_{j,\varepsilon }\cdot \nabla u_{j,\varepsilon }^Tdx +a\int _{\mathbb {R}^3}|u_{j,\varepsilon }^T|^{2k-2}|\nabla u_{j,\varepsilon }|^2dx\nonumber \\&=a(2k-2)\int _{|u_{j,\varepsilon }(x)|\le T}|u_{j,\varepsilon }^T|^{2k-2}| \nabla u_{j,\varepsilon }|^2dx+a\int _{\mathbb {R}^3}|u_{j,\varepsilon }^T|^{2k-2}|\nabla u_{j,\varepsilon }|^2dx\nonumber \\&\ge a\int _{\mathbb {R}^3}|u_{j,\varepsilon }^T|^{2k-2}|\nabla u_{j,\varepsilon }|^2dx\nonumber \\&\ge \frac{a}{k^2}\int _{\mathbb {R}^3}|\nabla (u_{j,\varepsilon }|u_{j,\varepsilon }^T|^{k-1})|^2dx\nonumber \\&\ge \frac{C}{k^2}\Big (\int _{\mathbb {R}^3}(u_{j,\varepsilon } |u_{j,\varepsilon }^T|^{k-1})^6dx\Big )^{\frac{1}{3}}. \end{aligned}$$

(3.4)

Consequently, by (3.2)–(3.4) we deduce that

$$\begin{aligned} \Big (\int _{\mathbb {R}^3}(u_{j,\varepsilon }|u_{j,\varepsilon }^T|^{k-1})^6dx\Big )^{\frac{1}{3}} \le Ck^2\Big (\int _{\mathbb {R}^3}(u_{j,\varepsilon }|u_{j,\varepsilon }^T|^{k-1})^{6\cdot \frac{2}{8-p}}dx\Big )^{\frac{8-p}{6}}. \end{aligned}$$

(3.5)

Suppose $\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{\frac{12k}{8-p}}dx<+\infty$. Let $T \rightarrow +\infty$ in (3.5) we have

$$\begin{aligned} \Big (\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{6k}dx\Big )^{\frac{1}{3}} \le Ck^2\Big (\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{6k\cdot \frac{2}{8-p}}dx\Big )^{\frac{8-p}{6}}. \end{aligned}$$

(3.6)

Set $\chi =\frac{2}{8-p}<1$. Starting from $k_1=\frac{8-p}{2}>1$, then one has

$$\begin{aligned} \Vert u_{j,\varepsilon }\Vert _{6k_1}\le (C k_1)^{\frac{1}{k_1}}\Vert u_{j,\varepsilon }\Vert _{6k_1\chi }=(C k_1)^{\frac{1}{k_1}}\Vert u_{j,\varepsilon }\Vert _6. \end{aligned}$$

Inductively, we choose $k_{i+1}=\frac{k_i}{\chi }$, $i=1,2,\ldots$. Then

$$\begin{aligned} \Vert u_{j,\varepsilon }\Vert _{6k_{i+1}}\le & {} (C k_{i+1})^{\frac{1}{k_{i+1}}}\Vert u_{j,\varepsilon }\Vert _{6k_{i+1}\chi }=(C k_{i+1})^{\frac{1}{k_{i+1}}}\Vert u_{j,\varepsilon }\Vert _{6k_i}\\\le & {} (C k_{i+1})^{\frac{1}{k_{i+1}}}(C k_i)^{\frac{1}{k_i}}\Vert u_{j,\varepsilon }\Vert _{6k_i\chi }=(C k_{i+1})^{\frac{1}{k_{i+1}}}(C k_i)^{\frac{1}{k_i}}\Vert u_{j,\varepsilon }\Vert _{6k_{i-1}}\\\le & {} \cdots \le (C k_{i+1})^{\frac{1}{k_{i+1}}}(C k_i)^{\frac{1}{k_i}}\cdots (C k_1)^{\frac{1}{k_1}} \Vert u_{j,\varepsilon }\Vert _{6k_1\chi }\\= & {} \prod \nolimits _{m=1}^{i+1}(Ck_m)^{\frac{1}{k_m}}\Vert u_{j,\varepsilon }\Vert _6. \end{aligned}$$

It is easy to see that

$$\begin{aligned} \prod \nolimits _{m=1}^{i+1}(Ck_m)^{\frac{1}{k_m}}=\exp \left( \sum \nolimits _{m=1}^{i+1}\frac{1}{k_m} \ln (Ck_m)\right) \end{aligned}$$

is convergent as $i\rightarrow +\infty$. Let $C_i=\prod _{m=1}^{i+1}(Ck_m)^{\frac{1}{k_m}}$. Then $C_i \rightarrow C_{\infty }$ as $i\rightarrow +\infty$. Hence, by $\Vert u_{j,\varepsilon }\Vert _{6k_1^{i+1}}\le C_i\Vert u_{j,\varepsilon }\Vert _6$ we obtain $\Vert u_{j,\varepsilon }\Vert _{L^{\infty }(\mathbb {R}^3)}\le C_{\infty }\Vert u_{j,\varepsilon }\Vert _6\le C$.

For $x_0 \in \mathbb {R}^3$, $0<\rho <R\le 1$, let $\eta \in C_0^{\infty }(\mathbb {R}^3, [0,1])$ be such that $\eta (x)=1$ for $x \in B_\rho (x_0)$, $\eta (x)=0$ for $x \not \in B_R(x_0)$ and $|\nabla \eta |\le \frac{C}{R-\rho }$. Take $\varphi =u_{j,\varepsilon }|u_{j,\varepsilon }|^{2k-2}\eta ^2 \ (k\ge 1)$ be a test function of (2.1), then corresponding formula (3.2) holds. Clearly,

$$\begin{aligned} \int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{p-2}u_{j,\varepsilon } \varphi dx =\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{p-2}|u_{j,\varepsilon }|^2|u_{j,\varepsilon }|^{2k-2}\eta ^2dx\le C\int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx \end{aligned}$$

and

$$\begin{aligned}&\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla \varphi dx=\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla [u_{j,\varepsilon }|u_{j,\varepsilon }|^{2k-2}\eta ^2]dx\\&\quad =(2k-1)\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k-2}\eta ^2|\nabla u_{j,\varepsilon }|^2dx+2\int _{\mathbb {R}^3}\eta u_{j,\varepsilon }|u_{j,\varepsilon }|^{2k-2}\nabla u_{j,\varepsilon }\cdot \nabla \eta dx\\&\quad \ge (2k-1)\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k-2}\eta ^2|\nabla u_{j,\varepsilon }|^2dx-2\delta \int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2|u_{j,\varepsilon }|^{2k-2} \eta ^2dx\\&\qquad -2C_\delta \int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k}|\nabla \eta |^2dx\\&\quad \ge \frac{2k-1}{2}\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k-2}\eta ^2|\nabla u_{j,\varepsilon }|^2dx-\frac{2C_\delta }{(R-\rho )^2}\int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx. \end{aligned}$$

Hence,

$$\begin{aligned}&a\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla \varphi dx+ b\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx \int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }\cdot \nabla \varphi dx\\&=(2k-1)(a+b\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx)\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k-2}\eta ^2|\nabla u_{j,\varepsilon }|^2dx\\&\quad +2(a+b\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2dx)\int _{\mathbb {R}^3}\eta u_{j,\varepsilon }|u_{j,\varepsilon }|^{2k-2}\nabla u_{j,\varepsilon }\cdot \nabla \eta dx\\&\ge (2k-1)a\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k-2}\eta ^2|\nabla u_{j,\varepsilon }|^2dx-2a\Big (\delta \int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }|^2|u_{j,\varepsilon }|^{2k-2}\eta ^2dx\\&\quad +C_\delta \int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k}|\nabla \eta |^2dx\Big )\\&\ge \frac{2k-1}{2}a\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k-2}\eta ^2|\nabla u_{j,\varepsilon }|^2dx-2C_\delta a\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k}|\nabla \eta |^2dx\\&=\frac{2k-1}{2}a\Big [\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k-2}\eta ^2|\nabla u_{j,\varepsilon }|^2dx+\frac{1}{k^2}\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k}|\nabla \eta |^2dx\Big ]\\&\quad -\frac{2k-1}{2}a\frac{1}{k^2}\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k}|\nabla \eta |^2dx-2C_\delta a\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k}|\nabla \eta |^2dx\\&\ge \frac{2k-1}{4k^2}a\int _{\mathbb {R}^3}|\nabla (|u_{j,\varepsilon }|^k\eta )|^2dx -\left( \frac{2k-1}{2}\frac{1}{k^2}+2C_\delta \right) a\int _{\mathbb {R}^3}|u_{j,\varepsilon }|^{2k}|\nabla \eta |^2dx\\&\ge \frac{2k-1}{4k^2}a\int _{\mathbb {R}^3}|\nabla (|u_{j,\varepsilon }|^k\eta )|^2dx -\left( \frac{2k-1}{2}\frac{1}{k^2}+2C_\delta \right) a\frac{C}{(R-\rho )^2}\int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx\\&\ge C\frac{2k-1}{4k^2}a\Big (\int _{\mathbb {R}^3}\Big ||u_{j,\varepsilon }|^k\eta \Big |^6dx\Big )^{\frac{1}{3}} -\left( \frac{2k-1}{2}\frac{1}{k^2}+2C_\delta \right) a\frac{C}{(R-\rho )^2}\int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx\\&\ge C\frac{2k-1}{4k^2}a\Big (\int _{B_\rho (x_0)}|u_{j,\varepsilon }|^{6k}dx\Big )^{\frac{1}{3}} -\left( \frac{2k-1}{2}\frac{1}{k^2}+2C_\delta \right) a\frac{C}{(R-\rho )^2}\int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned}&C\frac{2k-1}{4k^2}a\Big (\int _{B_\rho (x_0)}|u_{j,\varepsilon }|^{6k}dx\Big )^{\frac{1}{3}} -\left( \frac{2k-1}{2}\frac{1}{k^2}+2C_\delta \right) a\frac{C}{(R-\rho )^2} \int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx\\&\quad \le C\int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx, \end{aligned} \end{aligned}$$

then,

$$\begin{aligned} \begin{aligned}&C\frac{2k-1}{4k^2}a\Big (\int _{B_\rho (x_0)}|u_{j,\varepsilon }|^{6k}dx\Big )^{\frac{1}{3}} \le \left( \frac{2k-1}{2}\frac{1}{k^2}+2C_\delta \right) a\frac{C}{(R-\rho )^2} \int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx\\&\qquad +C\int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx, \end{aligned} \end{aligned}$$

which gives that

$$\begin{aligned} \Big (\int _{B_\rho (x_0)}|u_{j,\varepsilon }|^{6k}dx\Big )^{\frac{1}{3}}\le \frac{Ck^2}{(R-\rho )^2}\int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx, \end{aligned}$$

then,

$$\begin{aligned} \Big (\int _{B_\rho (x_0)}|u_{j,\varepsilon }|^{6k}dx\Big )^{\frac{1}{6k}} \le \left[ \frac{Ck^2}{(R-\rho )^2}\right] ^{\frac{1}{2k}} \Big (\int _{B_R(x_0)}|u_{j,\varepsilon }|^{2k}dx\Big )^{\frac{1}{2k}}. \end{aligned}$$

Similar to the proof of $\Vert u_{j,\varepsilon }\Vert _{L^{\infty }(\mathbb {R}^3)}\le C$ we can deduce that there exists a constant $C>0$ independent of R and $x_0$ such that

$$\begin{aligned} \Vert u_{j,\varepsilon }\Vert _{L^{\infty }(B_{\frac{R}{2}}(x_0))}\le C\Vert u_{j,\varepsilon }\Vert _{L^2(B_R(x_0))}. \end{aligned}$$

Consequently, for any $x \in \mathbb {R}^3\backslash (\mathcal {D}_\varepsilon )^\delta$, there exists $R_x>0$ such that $B_{R_x}(x)\subset \mathbb {R}^3\backslash (\mathcal {D}_\varepsilon )^\delta$, and so by (3.1) we have

$$\begin{aligned} \begin{aligned} |u_{j,\varepsilon }(x)|&\le \Vert u_{j,\varepsilon }\Vert _{L^{\infty }(B_{\frac{R_x}{2}}(x))}\le C\Vert u_{j,\varepsilon }\Vert _{L^2(B_{R_x}(x))}\\&\le C\Big (\int _{\mathbb {R}^3\backslash (\mathcal {D}_\varepsilon )^\delta }|u_{j,\varepsilon }|^2dx\Big )^{\frac{1}{2}}\le C(k,\delta )\varepsilon ^3, \end{aligned} \end{aligned}$$

which yields that the conclusion holds. This completes the proof. $\square$

According to [9], we have the following result.

Lemma 3.3

Let $\zeta >0$, $\{y_\varepsilon \}\subset \mathbb {R}^3$ and $\{u_\varepsilon \} \subset H^1(\mathbb {R}^3)\cap L^{\infty }(\mathbb {R}^3)$ be such that

$$\begin{aligned} \sup \limits _{\varepsilon >0}\Vert u_\varepsilon \Vert <+\infty , \ \int _{B_1(y_\varepsilon )}|u_\varepsilon |^2dx\ge \zeta \end{aligned}$$

and

$$\begin{aligned} \sup \{\langle J_\varepsilon ^\prime (u_\varepsilon ),\varphi \rangle :\varphi \in H_0^1(\mathcal {D}_\varepsilon ), \Vert \varphi \Vert _{H_0^1(\mathcal {D}_\varepsilon )}\le 1\}\rightarrow 0 \end{aligned}$$

as $\varepsilon \rightarrow 0$. Moreover, for any $\delta >0$, $\lim \nolimits _{\varepsilon \rightarrow 0}\Vert u_\varepsilon \Vert _{L^{\infty }(\mathbb {R}^3 \backslash (\mathcal {D}_\varepsilon )^\delta )}=0$. Then,

$$\begin{aligned} y_\varepsilon \in \mathcal {D}_\varepsilon \ \hbox {and} \ \lim \nolimits _{\varepsilon \rightarrow 0}\hbox {dist}(y_\varepsilon , \partial \mathcal {D}_\varepsilon )=+\infty . \end{aligned}$$

By Lemmas 3.1–3.3, with a similar argument in [9] we have the following lemma.

Lemma 3.4

Let $u_{j,\varepsilon } \rightharpoonup \tilde{u}_0$ in $H^1(\mathbb {R}^3)$ as $\varepsilon \rightarrow 0$. If $\liminf \nolimits _{\varepsilon \rightarrow 0}\Vert u_{j,\varepsilon }-\tilde{u}_0\Vert _p>0$, then there exist $m_j \in \mathbb {N}$, $m_j$ nonzero functions $\tilde{u}_i$ in $H^1(\mathbb {R}^3)$, $1 \le i \le m_j$, and $m_j$ sequences $\{y_{j,\varepsilon }^i\} \subset (\mathcal {D}_\varepsilon )^1$, $1 \le i \le m_j$ such that

$\mathrm{(i)}$ for any $1\le i \le m_j$, $\lim \nolimits _{\varepsilon \rightarrow 0}|y_{j,\varepsilon }^i|=+\infty$ and $\lim \nolimits _{\varepsilon \rightarrow 0}\hbox {dist}(y_{j,\varepsilon }^i,\partial \mathcal {D}_\varepsilon )=+\infty$. Moreover, if $i_1 \not =i_2$, $\lim \nolimits _{\varepsilon \rightarrow 0}|y_{j,\varepsilon }^{i_1}-y_{j,\varepsilon }^{i_2}|=+\infty$.

$\mathrm{(ii)}$ $\tilde{u}_0$ is a solution of

$$\begin{aligned} -(a+b\int _{\mathbb {R}^3}|\nabla u|^2dx)\triangle u+V(0)u=|u|^{p-2}u, \ u \in H^1(\mathbb {R}^3), \end{aligned}$$

and for any $1 \le i \le m_j$, $\tilde{u}_i$ is a nontrivial solution of

$$\begin{aligned} -(a+b\int _{\mathbb {R}^3}|\nabla u|^2dx)\triangle u+V(y_j^i)u=|u|^{p-2}u, \ u \in H^1(\mathbb {R}^3), \end{aligned}$$

where $y_j^i=\lim \nolimits _{\varepsilon \rightarrow 0}\varepsilon y_{j,\varepsilon }^i \in \overline{\mathcal {D}}$.

$\mathrm{(iii)}$ for any $2<t<6$, $\lim \nolimits _{\varepsilon \rightarrow 0}\Vert u_{j,\varepsilon }-\tilde{u}_0-\sum _{i=1}^{m_j}\tilde{u}_i(\cdot -y_{j,\varepsilon }^i)\Vert _t=0$.

For any $\varepsilon >0$ and any $1 \le j \le k$, define $y_{j,\varepsilon }^0=0$. Let $\varepsilon _n>0$ be such that $\varepsilon _n \rightarrow 0$ as $n\rightarrow \infty$. Up to a subsequence, we can assume that for every i, $\lim \nolimits _{n\rightarrow \infty }\varepsilon _n y_{j,\varepsilon _n}^i$ exists. Define the set of these limiting points by

$$\begin{aligned} \{x_1^*, \ldots , x_{s_j}^*\}=\{\lim \nolimits _{n\rightarrow \infty }\varepsilon _n y_{j,\varepsilon _n}^i: 0\le i \le m_j\}\subset \overline{\mathcal {D}} \end{aligned}$$

for some $1 \le s_j \le m_j$. Let

$$\begin{aligned} \begin{array}{ll} \theta _*= \left\{ \begin{array}{ll} \frac{1}{100}\hbox {min}\{|x_s^*-x_{s^\prime }^*|:1\le s<s^\prime \le s_j\}, &{} s_j \ge 2,\\ +\infty , &{} s_j=1. \end{array} \right. \end{array} \end{aligned}$$

(3.7)

Lemma 3.5

If $0<\delta <\theta _*$, then there exist $C>0$ and $c>0$ independent of n such that, for every $0\le i \le m_j$,

$$\begin{aligned} |\nabla u_{j,\varepsilon _n}(x)|+|u_{j,\varepsilon _n}(x)|\le Ce^{-c \varepsilon _n^{-1}} \end{aligned}$$

for large n, $\forall x \in \partial B(y_{j,\varepsilon _n}^i,\delta \varepsilon _n^{-1})$.

Proof

Arguing as in the proof of Lemma 4.5 in [9], it suffices to note that Eq. (4.40) in [9] turns into the following expression

$$\begin{aligned} \begin{aligned}&(a+b\int _{R_{m-1}}|\nabla u_{j,\varepsilon _n}|^2dx)\int _{R_{m-1}}|\nabla u_{j,\varepsilon _n}|^2\psi _m^2dx+\int _{R_{m-1}}V(\varepsilon _n x)|u_{j,\varepsilon _n}|^2\psi _m^2dx\\&\quad +\xi _n\int _{R_{m-1}}\chi _{\varepsilon _n}|u_{j,\varepsilon _n}|^2\psi _m^2dx -\int _{R_{m-1}}|u_{j,\varepsilon _n}|^p\psi _m^2dx\\&\qquad =-2(a+b\int _{R_{m-1}}|\nabla u_{j,\varepsilon _n}|^2dx)\int _{R_{m-1}}\psi _m u_{j,\varepsilon _n}\nabla u_{j,\varepsilon _n} \cdot \nabla \psi _mdx, \end{aligned} \end{aligned}$$

where $\psi _m(x)=\zeta _m(|x-y_{j,\varepsilon _n}^i|)$, $\zeta _m$ is a cut-off function satisfying that $0\le \zeta _m(t)\le 1$ for all $t \in \mathbb {R}$, $\zeta _m(t)=0$ for $t \le \frac{1}{2}\delta \varepsilon _n^{-1}+m-1$ or $t\ge \frac{3}{2}\delta \varepsilon _n^{-1}-m+1$, $\zeta _m(t)=1$ for $\frac{1}{2}\delta \varepsilon _n^{-1}+m\le t \le \frac{3}{2}\delta \varepsilon _n^{-1}-m$, $|\zeta _m^\prime (t)|\le 4$ for all $t \in \mathbb {R}$, $R_m:=\overline{B_{\frac{3}{2}\delta \varepsilon _n^{-1}-m}(y_{j,\varepsilon _n}^i)}\backslash B_{\frac{1}{2}\delta \varepsilon _n^{-1}+m}(y_{j,\varepsilon _n}^i)$. This completes the proof. $\square$

Lemma 3.6

For any $0 \le i \le m_j$, $\lim \limits _{\varepsilon \rightarrow 0}\hbox {dist}(\varepsilon y_{j,\varepsilon }^i, \mathcal {A})=0$.

Proof

It is easy to see that the conclusion holds for $i=0$. So it suffices to prove the case about $i \in [1,m_j]$. Supposing the conclusion is false, then there exists $i_0 \in [1,m_j]$, $\delta _0>0$ and $\varepsilon _n\rightarrow 0^+$ as $n\rightarrow \infty$ such that $\hbox {dist}(\varepsilon _n y_{j,\varepsilon _n}^{i_0},\mathcal {A})\ge \delta _0$. Without loss of generality, we can assume that for any i, $\lim \nolimits _{n\rightarrow \infty }\varepsilon _n y_{j,\varepsilon _n}^i$ exists. By $(V_2)$, there exists $\delta _1>0$ such that, for any $y \in \mathcal {D}^{\delta _1}$,

$$\begin{aligned} \inf \nolimits _{x \in B(y,\delta _1)\backslash \mathcal {D}}\nabla V(y)\cdot \nabla \hbox {dist}(x,\partial \mathcal {D})>0. \end{aligned}$$

(3.8)

By the fact that $y_j^{i_0}:=\lim \nolimits _{n\rightarrow \infty }\varepsilon _n y_{j,\varepsilon _n}^{i_0} \not \in \mathcal {A}$, there exists $\delta _2>0$ such that

$$\begin{aligned} \inf \limits _{x \in B(y_{j,\varepsilon _n}^{i_0},\delta _2\varepsilon _n^{-1})}\nabla V(\varepsilon _n x)\cdot \nabla V(\varepsilon _n y_{j,\varepsilon _n}^{i_0})\ge \frac{1}{2}|\nabla V(y_j^{i_0})|^2>0 \end{aligned}$$

(3.9)

for large n. Set

$$\begin{aligned} 0<\delta _0<\min \{\delta _1,\delta _2,\theta _*\}, \end{aligned}$$

where $\theta _*$ comes from (3.7). For notational simplicity, let $w_n=u_{j,\varepsilon _n}$. Noticing that $0<\delta _0<\theta _*$, Lemma 3.5 implies that there exist two constants $C>0$ and $c>0$ independent of n such that

$$\begin{aligned} |\nabla w_n(x)|+|w_n(x)|\le Ce^{-c\varepsilon _n^{-1}}, \ \forall x \in \partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1}) \end{aligned}$$

(3.10)

for large n. By virtue of Lemma 3.1, there exists $C>0$ independent of n such that

$$\begin{aligned} 0\le \xi _n:=\xi (u_{j,\varepsilon _n})= & {} \left( \int _{\mathbb {R}^3}\chi _\varepsilon (x) u_{j,\varepsilon _n}^2dx-1\right) _+^{\beta -1}\nonumber \\= & {} \left( \int _{\mathbb {R}^3}\chi _\varepsilon (x)w_n^2dx-1\right) _+^{\beta -1}\le C, \ \forall n \in \mathbb {N}. \end{aligned}$$

(3.11)

Set

$$\begin{aligned} \vec {t}_n=\nabla V(\varepsilon _n y_{j,\varepsilon _n}^{i_0}). \end{aligned}$$

By the fact that $w_n$ is a solution of (2.1) and the coefficients of (2.1) are all of class $C^1$, we know that $w_n$ is a $C^2$ function. Let $\nu$ be the unit outward normal to the boundary of $B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})$. Multiplying both sides of (2.1) by $\vec {t}_n \cdot \nabla w_n$ and integrating on $B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})$, we obtain

$$\begin{aligned} \begin{aligned}&a\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}-\Delta w_n(\vec {t}_n \cdot \nabla w_n)dx+b\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|\nabla w_n|^2dx\\&\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}-\Delta w_n(\vec {t}_n \cdot \nabla w_n)dx +\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}V(\varepsilon _n x) w_n(\vec {t}_n \cdot \nabla w_n)dx\\&\qquad +\xi _n\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\chi _{\varepsilon _n}(x)w_n(\vec {t}_n \cdot \nabla w_n)dx\\&\quad =\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})} |w_n|^{p-2}w_n(\vec {t}_n \cdot \nabla w_n)dx. \end{aligned} \end{aligned}$$

(3.12)

Integrating by parts, we see that

$$\begin{aligned} \begin{aligned}&\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}-\Delta w_n(\vec {t}_n \cdot \nabla w_n)dx\\&\quad =\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\nabla w_n\cdot \nabla (\vec {t}_n \cdot \nabla w_n)dx-\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}(\nabla w_n \cdot \nu )(\vec {t}_n \cdot \nabla w_n)dx. \end{aligned} \end{aligned}$$

Since $\vec {t}_n=\nabla V(\varepsilon _n y_{j,\varepsilon _n}^{i_0})$ does not depend on x, by virtue of Divergence theorem we obtain

$$\begin{aligned} \begin{aligned}&\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\nabla w_n\cdot \nabla (\vec {t}_n \cdot \nabla w_n)dx=\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\nabla w_n\cdot (\nabla ^2w_n \cdot \vec {t}_n)dx\\&\quad =\frac{1}{2}\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\nabla |\nabla w_n|^2\cdot \vec {t}_ndx\\&\quad =\frac{1}{2}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|\nabla w_n|^2(\vec {t}_n\cdot \nu )dx, \end{aligned} \end{aligned}$$

and so

$$\begin{aligned} \begin{aligned}&\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}-\Delta w_n(\vec {t}_n \cdot \nabla w_n)dx\\&\quad =\frac{1}{2}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|\nabla w_n|^2(\vec {t}_n\cdot \nu )dx-\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}(\nabla w_n \cdot \nu )(\vec {t}_n \cdot \nabla w_n)dx. \end{aligned} \end{aligned}$$

(3.13)

Similarly,

$$\begin{aligned} \begin{aligned}&\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\frac{1}{p}|w_n|^p(\nu \cdot \vec {t}_n)d x=\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\nabla (\frac{1}{p}|w_n|^p)\cdot \vec {t}_ndx\\&\quad =\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|w_n|^{p-2}w_n(\vec {t}_n\cdot \nabla w_n)dx, \end{aligned} \end{aligned}$$

so

$$\begin{aligned} \int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|w_n|^{p-2}w_n(\vec {t}_n \cdot \nabla w_n)dx=\frac{1}{p}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|w_n|^p(\vec {t}_n \cdot \nu )dx \end{aligned}$$

(3.14)

and

$$\begin{aligned} \begin{aligned}&\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}V(\varepsilon _n x) w_n(\vec {t}_n \cdot \nabla w_n)dx+\xi _n\int _{B(y_{j,\varepsilon _n}^{i_0}, \delta _0\varepsilon _n^{-1})}\chi _{\varepsilon _n}(x)w_n(\vec {t}_n \cdot \nabla w_n)dx\\&\quad =\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}[V(\varepsilon _n x) w_n+\xi _n\chi _{\varepsilon _n}(x)w_n](\vec {t}_n \cdot \nabla w_n)dx\\&\quad =\frac{1}{2}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}[V(\varepsilon _n x)+\xi _n\chi _{\varepsilon _n}(x)]w_n^2(\vec {t}_n \cdot \nu )dx\\&\qquad -\frac{1}{2}\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\Big ([\varepsilon _n \nabla V(\varepsilon _n x)+\xi _n\nabla \chi _{\varepsilon _n}]\cdot \vec {t}_n\Big ) w_n^2dx. \end{aligned} \end{aligned}$$

(3.15)

Substituting (3.13)–(3.15) into (3.12), we obtain the following local Pohozaev type identity

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\Big ([\varepsilon _n \nabla V(\varepsilon _n x)+\xi _n\nabla \chi _{\varepsilon _n}]\cdot \vec {t}_n\Big ) w_n^2dx\\&\quad =(a+b\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|\nabla w_n|^2dx)\Big [\frac{1}{2}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|\nabla w_n|^2(\vec {t}_n\cdot \nu )dx\\&\qquad -\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}(\nabla w_n \cdot \nu )(\vec {t}_n \cdot \nabla w_n)dx\Big ]\\&\qquad +\frac{1}{2}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}[V(\varepsilon _n x)+\xi _n\chi _{\varepsilon _n}]w_n^2(\vec {t}_n \cdot \nu )dx\\&\qquad -\frac{1}{p}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|w_n|^p(\vec {t}_n\cdot \nu )dx. \end{aligned} \end{aligned}$$

(3.16)

By the proof of Lemma 3.4 we can infer that $w_n(\cdot +y_{j,\varepsilon _n}^{i_0})\rightharpoonup \tilde{u}_{i_0} \not =0$ in $H^1(\mathbb {R}^3)$. Hence, by (3.9) we get that

$$\begin{aligned} \begin{aligned}&\varepsilon _n\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}[ \nabla V(\varepsilon _n x)\cdot \vec {t}_n]w_n^2dx\ge \frac{1}{2}\varepsilon _n|\nabla V(y_j^{i_0})|^2\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}w_n^2dx\\&\quad =\frac{1}{2}\varepsilon _n|\nabla V(y_j^{i_0})|^2\int _{B(0,\delta _0\varepsilon _n^{-1})}w_n^2(x+y_{j,\varepsilon _n}^{i_0})dx\\&\quad =\frac{1}{2}\varepsilon _n|\nabla V(y_j^{i_0})|^2[\int _{\mathbb {R}^3}\tilde{u}_{i_0}^2dx+o(1)]\\&\quad \ge \frac{1}{4}\varepsilon _n|\nabla V(y_j^{i_0})|^2\int _{\mathbb {R}^3}\tilde{u}_{i_0}^2dx \end{aligned} \end{aligned}$$

for large n. By (3.8), for any $x \in B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})\backslash \mathcal {D}_{\varepsilon _n}$, we have

$$\begin{aligned} \nabla V(\varepsilon _n y_{j,\varepsilon _n}^{i_0})\cdot \nabla \hbox {dist}(x,\partial \mathcal {D}_{\varepsilon _n})>0, \end{aligned}$$

which yields that

$$\begin{aligned} \vec {t}_n \cdot \nabla \chi _{\varepsilon _n}(x)\ge 0, \ \forall x \in B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1}) \end{aligned}$$

and so

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}\Big ([\varepsilon _n \nabla V(\varepsilon _n x)+\xi _n\nabla \chi _{\varepsilon _n}]\cdot \vec {t}_n\Big ) w_n^2dx\ge \frac{1}{8}\varepsilon _n|\nabla V(y_j^{i_0})|^2\int _{\mathbb {R}^3}\tilde{u}_{i_0}^2dx \end{aligned} \end{aligned}$$

for large n. Taking into account (3.10) and (3.11), we can see that

$$\begin{aligned} \begin{aligned}&(a+b\int _{B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|\nabla w_n|^2dx)\Big [\frac{1}{2}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|\nabla w_n|^2(\vec {t}_n\cdot \nu )dx\\&\quad -\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}(\nabla w_n \cdot \nu )(\vec {t}_n \cdot \nabla w_n)dx\Big ]\\&\quad +\frac{1}{2}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}[V(\varepsilon _n x)+\xi _n\chi _{\varepsilon _n}]w_n^2(\vec {t}_n \cdot \nu )dx-\frac{1}{p}\int _{\partial B(y_{j,\varepsilon _n}^{i_0},\delta _0\varepsilon _n^{-1})}|w_n|^p(\vec {t}_n\cdot \nu )dx\\&\qquad \le Ce^{-c\varepsilon _n^{-1}}. \end{aligned} \end{aligned}$$

Consequently, (3.16) implies that

$$\begin{aligned} \frac{1}{8}\varepsilon _n|\nabla V(y_j^{i_0})|^2\int _{\mathbb {R}^3}\tilde{u}_{i_0}^2dx\le Ce^{-c\varepsilon _n^{-1}} \end{aligned}$$

for large n, a contradiction. This completes the proof. $\square$

Lemma 3.7

For any $\delta >0$, there exist $C=C(\delta ,k)>0$ and $c=c(\delta ,k)>0$ independent of $\varepsilon$ such that for any $1 \le j \le k$,

$$\begin{aligned} |u_{j,\varepsilon }(x)| \le Ce^{-c\hbox {dist}(x,(\mathcal {A}^\delta )_\varepsilon )}, \ x \in \mathbb {R}^3. \end{aligned}$$

Proof

By Lemmas 3.4–3.6, arguing as in the proof of Lemma 4.8 in [9], it suffices to notice that Eq. (4.55) in [9] turns into the following expression

$$\begin{aligned} \begin{aligned}&(a+b\int _{B_m}|\nabla u_{j,\varepsilon }|^2dx)\int _{B_m}|\nabla u_{j,\varepsilon }|^2\phi _m^2dx+\int _{B_m}V(\varepsilon x)|u_{j,\varepsilon }|^2\phi _m^2dx\\&\qquad +\xi _\varepsilon \int _{B_m}\chi _\varepsilon (x)|u_{j,\varepsilon }|^2\phi _m^2dx -\int _{B_m}|u_{j,\varepsilon }|^p\phi _m^2dx\\&\quad =-2(a+b\int _{B_m}|\nabla u_{j,\varepsilon }|^2dx)\int _{B_m}u_{j,\varepsilon }\phi _m\nabla u_{j,\varepsilon } \cdot \nabla \phi _mdx, \end{aligned} \end{aligned}$$

where $B_m=\{x \in \mathbb {R}^3: \hbox {dist}(x,\overline{(\mathcal {A}^\delta )_\varepsilon })\ge R_0+m-1\}$, $\phi _m(x)=\rho _m(\hbox {dist}(x,\overline{(\mathcal {A}^\delta )_\varepsilon }))$, $\rho _m$ is a cut-off function satisfying $0\le \rho _m(t)\le 1$ for all $t \in \mathbb {R}$, $\rho _m(t)=0$ for $t \le R_0+m-1$, $\rho _m(t)=1$ for $t \ge R_0+m$ and $|\rho ^\prime _m(t)|\le 4$ for all $t \in \mathbb {R}$. This completes the proof.

Since $\mathcal {A} \subset D$ is a compact set, $\hbox {dist}(\mathcal {A},\partial D)>0$. With the aid of Lemma 3.7, choosing $0<\delta <\hbox {dist}(\mathcal {A}, \partial D)$, one has

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0}\int _{\mathbb {R}^3}\chi _\varepsilon (x)u_{j,\varepsilon }^2dx=0, \ \forall 1\le j \le k. \end{aligned}$$

By the expression of $Q_\varepsilon (\cdot )$ we have $Q_\varepsilon (u_{j,\varepsilon })=0$ for $\varepsilon >0$ sufficiently small. Consequently, the critical point $u_{j,\varepsilon }$ of $J_\varepsilon$ established by Lemma 2.3 is a solution of (1.5), and the following lemma holds.

Lemma 3.8

There exists $\varepsilon _k>0$ such that if $0<\varepsilon <\varepsilon _k$, then for any $1 \le j\le k$, $u_{j,\varepsilon }$ is a solution of (1.5).

Proof of Theorem 1.1

By Lemmas 2.3, 3.7–3.8, we get the results of Theorem 1.1. $\square$

4 Proof of Theorem 1.2

In the following, we regard $b\in (0,1]$ in Eq. (1.1) as a parameter, then by the proof of Theorem 1.1, for any positive integer k, there exists $\varepsilon _k>0$ independent of $b_{n}$, such that for $0<\varepsilon <\varepsilon _k$, Eq. (1.1) possesses at least k pairs of sign-changing solutions $\pm u_{j,\varepsilon }^b$, $j=1,2,\ldots ,k$. From now on, we denote $I_\varepsilon$, $c_j^\varepsilon$ by $I_\varepsilon ^b$, $c_j^{\varepsilon ,b}$, respectively.

Proof of Theorem 1.2

By $b_n\rightarrow 0$, we can assume that $\{b_n\}\subset (0, 1)$, then by the proof of Theorem 1.1, for any positive integer k, there exists $\varepsilon _k>0$ independent of $b_{n}$, such that for $0<\varepsilon <\varepsilon _k$, Eq. (1.1) possesses at least k pairs of sign-changing solutions $\pm u_{j,\varepsilon }^{b_n}$, $j=1,2,\ldots ,k$. Furthermore, $I_\varepsilon ^{b_n}(u_{j,\varepsilon }^{b_n})\le J_\varepsilon ^{b_n}(u_{j,\varepsilon }^{b_n})=c_{j+1}^{\varepsilon ,b_n}\le \tilde{c}_{k+1}$ and $(I_\varepsilon ^{b_n})^\prime (u_{j,\varepsilon }^{b_n})=0$. Consequently, by $(V_1)$

$$\begin{aligned} \begin{aligned} \tilde{c}_{k+1}\ge I_\varepsilon ^{b_n}(u_{j,\varepsilon }^{b_n})&=I_\varepsilon ^{b_n}(u_{j,\varepsilon }^{b_n})-\frac{1}{p}\langle (I_\varepsilon ^{b_n})^\prime (u_{j,\varepsilon }^{b_n}),u_{j,\varepsilon }^{b_n} \rangle \\&=\left( \frac{1}{2}-\frac{1}{p}\right) a\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }^{b_n}|^2dx+\left( \frac{1}{4}-\frac{1}{p}\right) b_n\left( \int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }^{b_n}|^2dx\right) ^2\\&\quad +\left( \frac{1}{2}-\frac{1}{p}\right) \int _{\mathbb {R}^3}V(\varepsilon x)|u_{j,\varepsilon }^{b_n}|^2dx, \end{aligned} \end{aligned}$$

which means that $\{u_{j,\varepsilon }^{b_n}\}$ is bounded in $H^1(\mathbb {R}^3)$. Therefore, up to a subsequence, there exists $u_{j,\varepsilon }^0 \in H^1(\mathbb {R}^3)$ such that $u_{j,\varepsilon }^{b_n}\rightharpoonup u_{j,\varepsilon }^0$ in $H^1(\mathbb {R}^3)$ as $n\rightarrow \infty$. Define

$$\begin{aligned} I_\varepsilon ^0(u)=\frac{a}{2}\int _{\mathbb {R}^3}|\nabla u|^2dx +\frac{1}{2}\int _{\mathbb {R}^3}V(\varepsilon x)u^2dx-\frac{1}{p}\int _{\mathbb {R}^3}|u|^pdx, \end{aligned}$$

which is the energy functional of the following equation

$$\begin{aligned} -a\Delta u+V(\varepsilon x)u=|u|^{p-2}u, \ x \ \in \mathbb {R}^3. \end{aligned}$$

(4.1)

For any $\varphi \in H^1(\mathbb {R}^3)$, we have

$$\begin{aligned} \begin{aligned} 0=&\langle (I_\varepsilon ^{b_n})^\prime (u_{j,\varepsilon }^{b_n}),\varphi \rangle \\ =&a\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }^{b_n}\cdot \nabla \varphi dx+b_n\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }^{b_n}|^2dx\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }^{b_n}\cdot \nabla \varphi dx\\&+\int _{\mathbb {R}^3}V(\varepsilon x)u_{j,\varepsilon }^{b_n}\varphi dx-\int _{\mathbb {R}^3}|u_{j,\varepsilon }^{b_n}|^{p-2}u_{j,\varepsilon }^{b_n}\varphi dx. \end{aligned} \end{aligned}$$

(4.2)

Since $u_{j,\varepsilon }^{b_n}\rightharpoonup u_{j,\varepsilon }^0$ in $H^1(\mathbb {R}^3)$ as $n\rightarrow \infty$, by the definition of weak convergence we obtain that

$$\begin{aligned} \int _{\mathbb {R}^3}[a\nabla u_{j,\varepsilon }^{b_n}\cdot \nabla \varphi +V(\varepsilon x)u_{j,\varepsilon }^{b_n}\varphi ]dx\rightarrow \int _{\mathbb {R}^3}[a\nabla u_{j,\varepsilon }^0\cdot \nabla \varphi +V(\varepsilon x)u_{j,\varepsilon }^0\varphi ]dx \end{aligned}$$

(4.3)

as $n\rightarrow \infty$. Since $b_n\rightarrow 0$ as $n\rightarrow \infty$ and $\{||u_{j,\varepsilon }^{b_n}||\}$ is bounded, one has

$$\begin{aligned} b_n\int _{\mathbb {R}^3}|\nabla u_{j,\varepsilon }^{b_n}|^2dx\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }^{b_n}\cdot \nabla \varphi dx\rightarrow 0. \end{aligned}$$

(4.4)

By Young inequality we conclude that

$$\begin{aligned} \begin{aligned}&\Big |[|u_{j,\varepsilon }^{b_n}|^{p-2}u_{j,\varepsilon }^{b_n}-|u_{j,\varepsilon }^0|^{p-2} u_{j,\varepsilon }^0]\varphi \Big |\\&\quad \le |u_{j,\varepsilon }^{b_n}|^{p-1}|\varphi |+|u_{j,\varepsilon }^0|^{p-1}|\varphi |\\&\quad \le 2^{p-1} |u_{j,\varepsilon }^{b_n}-u_{j,\varepsilon }^0|^{p-1}|\varphi |+(2^{p-1}+1) |u_{j,\varepsilon }^0|^{p-1}|\varphi |\\&\quad \le \delta |u_{j,\varepsilon }^{b_n}-u_{j,\varepsilon }^0|^p+C_\delta |\varphi |^p+(2^{p-1}+1)| u_{j,\varepsilon }^0|^{p-1}|\varphi |. \end{aligned} \end{aligned}$$

Set

$$\begin{aligned} G_{\delta ,n}(x)=\max \left\{ \Big |[|u_{j,\varepsilon }^{b_n}|^{p-2} u_{j,\varepsilon }^{b_n}-|u_{j,\varepsilon }^0|^{p-2}u_{j,\varepsilon }^0]\varphi \Big | -\delta |u_{j,\varepsilon }^{b_n}-u_{j,\varepsilon }^0|^p,0\right\} . \end{aligned}$$

Then $0\le G_{\delta ,n}(x)\le C_\delta |\varphi |^p+(2^{p-1}+1)|u_{j,\varepsilon }^0|^{p-1}|\varphi | \in L^1(\mathbb {R}^3)$ and $G_{\delta ,n}(x)\rightarrow 0$ a.e. on $\mathbb {R}^3$. Hence, by Lebesgue dominated convergence theorem we get that $\int _{\mathbb {R}^3}G_{\delta ,n}(x)dx\rightarrow 0$ as $n\rightarrow \infty$. And so

$$\begin{aligned} \begin{aligned}&\limsup \limits _{n\rightarrow \infty }|\int _{\mathbb {R}^3}[|u_{j,\varepsilon }^{b_n}|^{p-2} u_{j,\varepsilon }^{b_n}-|u_{j,\varepsilon }^0|^{p-2}u_{j,\varepsilon }^0]\varphi dx|\\&\quad \le \limsup \limits _{n\rightarrow \infty }\int _{\mathbb {R}^3}G_{\delta ,n}(x)dx+\delta \limsup \limits _{n\rightarrow \infty }\int _{\mathbb {R}^3}|u_{j,\varepsilon }^{b_n}-u_{j,\varepsilon }^0|^pdx\\&\quad \le C\delta . \end{aligned} \end{aligned}$$

By the arbitrariness of $\delta$ we see that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\int _{\mathbb {R}^3} |u_{j,\varepsilon }^{b_n}|^{p-2}u_{j,\varepsilon }^{b_n}\varphi dx =\int _{\mathbb {R}^3}|u_{j,\varepsilon }^0|^{p-2}u_{j,\varepsilon }^0\varphi dx. \end{aligned}$$

(4.5)

Substituting (4.3)–(4.5) into (4.2), we derive that

$$\begin{aligned} a\int _{\mathbb {R}^3}\nabla u_{j,\varepsilon }^0\cdot \nabla \varphi dx +\int _{\mathbb {R}^3}V(\varepsilon x)u_{j,\varepsilon }^0\varphi dx=\int _{\mathbb {R}^3}|u_{j,\varepsilon }^0|^{p-2}u_{j,\varepsilon }^0\varphi dx, \end{aligned}$$

i.e., $\langle (I_\varepsilon ^0)^\prime (u_{j,\varepsilon }^0),\varphi \rangle =0$ for all $\varphi \in H^1(\mathbb {R}^3)$, which yields that $(I_\varepsilon ^0)^\prime (u_{j,\varepsilon }^0)=0$, that is, $u_{j,\varepsilon }^0$ is a weak solution of (4.1). This completes the proof. $\square$