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Multiple positive solutions to Kirchhoff equations with competing potential functions in \(\mathbb{R}^{3}\)

Abstract

In this paper, we study the existence of multiple positive solutions to the following Kirchhoff equation with competing potential functions:

$$ \textstyle\begin{cases} -(\varepsilon ^{2}a+\varepsilon b{\int _{\mathbb{R}^{3}}}{ \vert \nabla v \vert } ^{2})\Delta v+V(x)v=K(x) \vert v \vert ^{p-1}v \quad \mbox{in }\mathbb{R}^{3}, \\ v>0, \quad v\in H^{1}(\mathbb{R}^{3}), \end{cases} $$

where \(\varepsilon >0\) is a small parameter, \(a,b>0\) are constants, \(3< p<5\). We relate the number of solutions with the topology of the global minima set of the function \({V^{\frac{2}{p-1}-\frac{1}{2}}(x)}/ {K^{\frac{2}{p-1}}(x)}\). The Nehari manifold and Ljusternik–Schnirelmann category theory are applied in our study.

Introduction

In this paper, we study the existence of multiple positive solutions to the Kirchhoff equation with competing potential functions:

$$ \textstyle\begin{cases} -(\varepsilon ^{2}a+\varepsilon b{\int _{\mathbb{R}^{3}}}{ \vert \nabla v \vert } ^{2})\Delta v+V(x)v=K(x) \vert v \vert ^{p-1}v \quad \mbox{in }\mathbb{R}^{3}, \\ v>0, \quad v\in H^{1}(\mathbb{R}^{3}), \end{cases} $$
(1.1)

where \(\varepsilon >0\) is a small parameter, \(a,b>0\) are constants, \(3< p<5\), \(V(x)\) and \(K(x)\) are positive continuous functions satisfying

  1. (H)

    \(\inf_{x\in \mathbb{R}^{3}}V(x)=\bar{V}>0\), \(K(x)>0\) and \(K(x)\) is bounded.

In recent years, the elliptic Kirchhoff type equations have been studied extensively by many authors, and they are related to the stationary analogue of the equation

$$ u_{tt} - \biggl(a + b \int _{\varOmega } \vert \nabla u \vert ^{2} \biggr) \Delta u = g(x,t) $$
(1.2)

proposed by Kirchhoff [14] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations.

Some early classical studies of Kirchhoff equations were those by Bernstein [4] and Pohozaev [21]. Equation (1.2) received much attention after Lions [18] had proposed an abstract framework to the problem. In 2006, Perera and Zhang [20, 31] obtained existence and multiplicity of solutions via variational methods. Recently several interesting results can be found in Azzollini [2], Li et al. [15], Li et al. [16], Liang et al. [17], Wu [29], Zhang [30], etc.

On the other hand, the well-known Schrödinger equation

$$ -\varepsilon ^{2}\Delta v+V(x)v=f(v) \quad \mbox{in } \mathbb{R}^{N}, $$
(1.3)

has been paid much attention to after the celebrated work of Floer and Weinstein [11]. Many famous mathematicians have obtained a lot of interesting results, we only refer to [5, 6, 9, 22, 26, 27] and the references therein.

Recently, many authors have studied the existence and concentration behavior of positive solutions for Kirchhoff type equations in \(\mathbb{R}^{3}\). In [12], He and Zou studied (1.1) with subcritical nonlinearity. In [23], Sun and Zhang investigated the uniqueness of positive ground state solutions for Kirchhoff type equations with constant coefficients and then studied the existence and concentration behavior of Kirchhoff type problems in \(\mathbb{R}^{3}\) with competing potentials. For more interesting results, we refer to [10, 13, 24, 25] etc.

In [8], Cingolani and Lazzo studied the existence of multiple positive solutions to the nonlinear Schrödinger equation with competing potential functions

$$ -\varepsilon ^{2}\Delta v+V(x)v=K(x) \vert v \vert ^{p-2}v+Q(x) \vert u \vert ^{q-2}u \quad \mbox{in } \mathbb{R}^{N}. $$
(1.4)

They related the number of solutions with the topology of the global minima set of a suitable ground energy function. If \(Q(x)=0\) in (1.4), the ground energy function is \(V^{(2p+2N-Np)/(2p-4)}(x)/{K^{2/(p-2)}(x)}\).

Inspired by [8], we consider the existence of multiple positive solutions for the Kirchhoff equation (1.1) where a nonlocal term \({\int _{\mathbb{R}^{3}}}{|\nabla v|}^{2}\) appears in it. Because of the nonlocal term, the method of proof in [8] cannot work directly for our case, and several special difficulties would be faced. For example, we cannot get the Palais–Smale condition if we deal with it in a completely the same way as in [8], which forces us to develop new techniques to solve it. Moreover, the appearance of a competing potential function \(K(x)\) and the nonlocal term in (1.1) will bring troubles to the uniform estimate in Sect. 4.

Let us denote by M the global minima set of the function \(g(x):=\frac{V ^{\frac{2}{p-1}-\frac{1}{2}}(x)}{K^{\frac{2}{p-1}}(x)}\), i.e.,

$$ M=\Bigl\{ \xi \in \mathbb{R}^{3}: g(\xi )=\inf _{x \in \mathbb{R}^{3}}g(x)\Bigr\} . $$
(1.5)

We recall that, if Y is a closed subset of a topological space X, \(\mathrm{cat}_{X}Y\) denotes the Ljusternik–Schnirelmann category of Y in X, namely the least number of closed and contractible sets in X which cover Y. For \(\delta >0\), we denote

$$ M_{\delta }:=\bigl\{ x\in \mathbb{R}^{3}:\operatorname{dist}(x, M) \le \delta \bigr\} . $$

Our main result is the following.

Theorem 1.1

Suppose that (H) holds and

$$ \frac{\liminf \limits _{|x|\to \infty }V^{\frac{2}{p-1}-\frac{1}{2}}(x)}{ \limsup \limits _{|x|\to \infty }K^{\frac{2}{p-1}}(x)}> \inf \limits _{x\in \mathbb{R}^{3}}\frac{V^{\frac{2}{p-1}-\frac{1}{2}}(x)}{K ^{\frac{2}{p-1}}(x)}, $$
(1.6)

then, for any \(\delta >0\), there exists \(\varepsilon _{\delta }>0\) such that equation (1.1) has at least \(\mathrm{cat}_{M_{ \delta }}(M)\) solutions for \(\varepsilon \in (0, \varepsilon _{\delta })\).

Remark 1.2

By assumption (1.6), the set M defined in (1.5) is not empty and is a bounded closed set in \(\mathbb{R}^{3}\).

Remark 1.3

Consider the following Kirchhoff equation with constant coefficients:

$$ \textstyle\begin{cases} -(a+b{\int _{\mathbb{R}^{3}}}{ \vert \nabla v \vert }^{2})\Delta v+mv=n \vert v \vert ^{p-1}v \quad \mbox{in } \mathbb{R}^{3}, \\ v>0, \quad v\in H^{1}(\mathbb{R}^{3}), \end{cases} $$
(1.7)

where \(a,b>0\) are constants, \(3< p<5\), \(m,n>0\) are taken as variable parameters here. We define \(c(m,n)\) by

$$ c(m,n):=\inf_{v\in \mathcal{N}^{(m,n)}}I^{(m,n)}(v), $$

where \(I^{(m,n)}\) is the energy functional and \(\mathcal{N}^{(m,n)}\) is the Nehari manifold associated to (1.7), i.e.,

$$ I^{(m,n)}(v)=\frac{1}{2}{ \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla v \vert }^{2}+mv ^{2}}\bigr)+ \frac{b}{4}\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla v \vert }^{2} \biggr)^{2}- \frac{1}{p+1} \int _{\mathbb{R}^{3}}n \vert v \vert ^{p+1} $$

and

$$ \mathcal{N}^{(m,n)}=\biggl\{ v\in H^{1}\bigl( \mathbb{R}^{3}\bigr) \backslash \{0\}: { \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla v \vert }^{2}+mv^{2}\bigr)}+ b\biggl( \int _{\mathbb{R} ^{3}}{ \vert \nabla v \vert }^{2} \biggr)^{2}= \int _{\mathbb{R}^{3}}n \vert v \vert ^{p+1}\biggr\} . $$

Then, by Lemma 3.6 in [23], we know that \(c(m,n):\mathbb{R} ^{+}\times \mathbb{R}^{+}\to \mathbb{R}^{+}\) is continuous. Moreover, let \(m_{1},m_{2},n_{1},n_{2}>0\), then

$$ c(m_{1},n_{1})< c(m_{2},n_{2}) \quad \mbox{if and only if}\quad {m_{1}^{ \frac{2}{p-1}-\frac{1}{2}}}/{n_{1}^{\frac{2}{p-1}}}< {m_{2}^{ \frac{2}{p-1}-\frac{1}{2}}}/{n_{2}^{\frac{2}{p-1}}}. $$
(1.8)

Now we define the ground energy function \(G(\xi )\) which was first introduced in [27] by

$$ G(\xi ):=c\bigl(V(\xi ),K(\xi )\bigr)\quad \mbox{for }\xi \in \mathbb{R}^{3}. $$

Then by (1.8) we know that \(\xi \in \mathbb{R}^{3}\) satisfies \(G(\xi )=\inf_{s\in \mathbb{R}^{3}}G(s)\) if and only if \(\xi \in M\), where M is defined in (1.5). Now define \(c_{0}:=\inf_{s\in \mathbb{R}^{3}}G(s)\).

Let \(V_{\infty }\) and \(K_{\infty }\) be defined as

$$ V_{\infty }:=\liminf_{|x|\to \infty }V(x),\qquad K_{\infty }:=\limsup_{|x|\to \infty }K(x), $$

and let \(c_{\infty }:=c(V_{\infty },K_{\infty })\). If \(V_{\infty }=+ \infty \), define \(c_{\infty }:=+\infty \). Then, by (1.8), we get that condition (1.6) is equivalent to

$$ c_{0}< c_{\infty }. $$
(1.9)

Preliminaries

First let \(u(x)=v(\varepsilon x)\), then equation (1.1) becomes the following equivalent equation:

$$ -\biggl(a+b{ \int _{\mathbb{R}^{3}}} { \vert \nabla u \vert }^{2}\biggr) \Delta u+V(\varepsilon x)u=K( \varepsilon x) \vert u \vert ^{p-1}u \quad \mbox{in } \mathbb{R}^{3}. $$
(2.1)

Let \(E_{\varepsilon }:=\{u\in H^{1}(\mathbb{R}^{3}):{\int _{\mathbb{R} ^{3}}}{V(\varepsilon x)}u^{2}<+\infty \}\) be the Hilbert subspace of \(H^{1}(\mathbb{R}^{3})\) with the norm

$$ \Vert u \Vert _{E_{\varepsilon }}:=\biggl( \int _{\mathbb{R}^{3}}\bigl({a \vert \nabla u \vert }^{2}+ {V(\varepsilon x)}u^{2}\bigr)\biggr)^{1/2}. $$

Then a weak solution of problem (2.1) in \(E_{\varepsilon }\) is a critical point of the energy functional \(I_{\varepsilon }:E _{\varepsilon }\to \mathbb{R}\) given by

$$ I_{\varepsilon }(u):=\frac{1}{2}{ \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla u \vert } ^{2}+{V(\varepsilon x)}u^{2}}\bigr)+ \frac{b}{4}\biggl( \int _{\mathbb{R}^{3}} { \vert \nabla u \vert }^{2} \biggr)^{2}- \frac{1}{p+1} \int _{\mathbb{R}^{3}}K(\varepsilon x) \vert u \vert ^{p+1}. $$

Moreover, \(I_{\varepsilon }\in C^{1}(E_{\varepsilon },\mathbb{R}) \). We define the Nehari manifold for (2.1) by

$$ \mathcal{N}_{\varepsilon }:=\biggl\{ u\in E_{\varepsilon }\backslash \{0\}: { \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla u \vert }^{2}+{V( \varepsilon x)}u^{2}}\bigr)+ b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert }^{2} \biggr)^{2}= \int _{\mathbb{R}^{3}}K( \varepsilon x) \vert u \vert ^{p+1} \biggr\} . $$

That is,

$$ \mathcal{N}_{\varepsilon }=\bigl\{ u\in E_{\varepsilon }\backslash \{0\}: \bigl\langle I_{\varepsilon }'(u),u \bigr\rangle =0\bigr\} . $$

By [23], we have

Lemma 2.1

For any \(u\in E_{\varepsilon }\backslash \{0\}\), there exists unique \(t(u)>0\) such that \(t(u)u\in \mathcal{N}_{\varepsilon }\) and the maximum of \(I_{\varepsilon }(tu)\) for \(t\ge 0\) is achieved at \(t=t(u)\).

We denote by \(S(u):= \langle I_{\varepsilon }'(u),u \rangle \), then we have the following.

Lemma 2.2

For any \(\varepsilon >0\), there exist \(\sigma _{\varepsilon }, \tau _{\varepsilon }>0\) such that, for any \(u\in \mathcal{N}_{\varepsilon }\),

$$ \Vert u \Vert _{E_{\varepsilon }}\ge \sigma _{\varepsilon },\qquad \bigl\langle S'(u),u \bigr\rangle \le -\tau _{\varepsilon }. $$
(2.2)

Proof

Since the embedding \(E_{\varepsilon }\hookrightarrow L^{r}(\mathbb{R} ^{3})\) is continuous for \(2\le r \le 6\), then we have that, for any \(u\in \mathcal{N}_{\varepsilon }\),

$$ 0< \Vert u \Vert _{E_{\varepsilon }}^{2}\le \int _{\mathbb{R}^{3}}K(\varepsilon x) \vert u \vert ^{p+1}\le C_{1} \Vert K \Vert _{\infty } \Vert u \Vert _{E_{\varepsilon }}^{p+1}, $$

where \(C_{1}\) is a positive constant, and which implies the first inequality in (2.2). Furthermore,

$$\begin{aligned} \bigl\langle S'(u),u \bigr\rangle =&2 \Vert u \Vert _{E_{\varepsilon }}^{2}+4b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert }^{2} \biggr)^{2}- (p+1) \int _{\mathbb{R}^{3}}K( \varepsilon x) \vert u \vert ^{p+1} \\ =&-2 \Vert u \Vert _{E_{\varepsilon }}^{2}-(p-3) \int _{\mathbb{R}^{3}}K(\varepsilon x) \vert u \vert ^{p+1} \le -2 \Vert u \Vert _{E_{\varepsilon }}^{2} \le -2\sigma _{\varepsilon }=:-\tau _{\varepsilon }, \end{aligned}$$

which gives the second inequality in (2.2). □

By Lemma 2.2, we know that (see Chap. 6 of [1]) \(\mathcal{N}_{\varepsilon }\) is a \(C^{1}\) manifold of codimension one in \(E_{\varepsilon }\) and \(\mathcal{N}_{\varepsilon }\) is a natural constraint for \(I_{\varepsilon }\), i.e., if u is a critical point of \(I_{\varepsilon }|_{\mathcal{N}_{\varepsilon }}\) (\(I_{\varepsilon }\) constrained on \(\mathcal{N}_{\varepsilon }\)), then u is a weak solution of (2.1) in \(E_{\varepsilon }\).

Now define the ground energy \(c_{\varepsilon }\) of functional \(I_{\varepsilon }\) by \(c_{\varepsilon }:= \inf_{u\in \mathcal{N}_{\varepsilon }}I_{\varepsilon }(u)\). By Lemma 3.4 of [23], we know that there exists \(\bar{c}>0\) such that \(c_{\varepsilon }>\bar{c}\) for each \(\varepsilon >0\) and \(\limsup_{\varepsilon \to 0^{+}}c_{\varepsilon }\le c_{0}\), where \(c_{0}\) is defined in Remark 1.3. Moreover, we can prove the following.

Lemma 2.3

$$ \liminf_{\varepsilon \to 0^{+}}c_{\varepsilon }\ge c_{0}. $$

Proof

By [23], there exists a positive ground state solution \(u_{\varepsilon }\) of (2.1) which satisfies \(I_{\varepsilon }(u_{\varepsilon })=c_{\varepsilon }\) for sufficiently small \(\varepsilon >0\). Now, by contradiction, we assume that there exist \(d_{0}>0\) and a subsequence \(\{u_{\varepsilon _{k}}\}\) of \(\{u_{\varepsilon }\}\) such that \(c_{\varepsilon _{k}}=I_{\varepsilon _{k}}(u_{\varepsilon _{k}})\to c_{0}-d_{0}\), i.e.,

$$ \int _{\mathbb{R}^{3}}\biggl[\frac{1}{4}\bigl(a{ \vert \nabla u_{\varepsilon _{k}} \vert }^{2}+ {V(\varepsilon _{k} x)}u_{\varepsilon _{k}}^{2}\bigr)+ \biggl(\frac{1}{4}- \frac{1}{p+1}\biggr)K(\varepsilon _{k} x) \vert u_{\varepsilon _{k}} \vert ^{p+1}\biggr]\to c _{0}-d_{0}. $$
(2.3)

From [23], we know there exists \(\{y_{\varepsilon _{k}}\} \subset \mathbb{R}^{3}\) such that \(\varepsilon _{k}y_{\varepsilon _{k}} \to x_{0} \in M\) (defined in (1.5)), and if we let \(w_{k}(x):=u _{\varepsilon _{k}}(x+y_{\varepsilon _{k}})\), then \(w_{k}\to w_{0}\) in \(H^{1}(\mathbb{R}^{3})\), where \(w_{0}\) is the unique positive ground state solution of

$$ -\biggl(a+b{ \int _{\mathbb{R}^{3}}} { \vert \nabla w_{0} \vert }^{2}\biggr)\Delta w_{0}+V(x_{0})w _{0}=K(x_{0})w_{0}^{p}. $$

Then \(c_{0}=\int _{\mathbb{R}^{3}}[\frac{1}{4}(a{|\nabla w_{0}|}^{2}+ {V(x_{0})}w_{0}^{2})+ (\frac{1}{4}-\frac{1}{p+1})K(x_{0})|w_{0}|^{p+1}]\) and there exists \(\rho _{0}>0\) such that

$$ \int _{B_{\rho _{0}}(0)}\biggl[\frac{1}{4}\bigl(a{ \vert \nabla w_{0} \vert }^{2}+{V(x_{0})}w _{0}^{2}\bigr)+ \biggl(\frac{1}{4}-\frac{1}{p+1} \biggr)K(x_{0}) \vert w_{0} \vert ^{p+1} \biggr]>c_{0}- \frac{1}{3}d_{0}. $$
(2.4)

By (2.3) and \(w_{k}(x)=u_{\varepsilon _{k}}(x+y_{\varepsilon _{k}})\), we can choose large and fixed \(\rho _{1}>\rho _{0}\) such that

$$\begin{aligned} \begin{aligned} & \int _{B_{\rho _{1}(0)}}\biggl[\frac{1}{4}\bigl(a{ \vert \nabla w_{k} \vert }^{2}+{V( \varepsilon _{k} x+ \varepsilon _{k}y_{\varepsilon _{k}})}w_{k}^{2}\bigr) \\ &\quad {}+\biggl(\frac{1}{4}-\frac{1}{p+1}\biggr)K(\varepsilon _{k} x+\varepsilon _{k}y_{ \varepsilon _{k}}) \vert w_{k} \vert ^{p+1}\biggr] < c_{0}- \frac{2}{3}d_{0}. \end{aligned} \end{aligned}$$
(2.5)

Thus letting \(k\to \infty \) in (2.5), by \(\varepsilon _{k}y _{\varepsilon _{k}}\to x_{0}\) and \(w_{k}\to w_{0}\) in \(H^{1}\), we have

$$ \frac{1}{4} \int _{B_{\rho _{1}}(0)}\bigl(a{ \vert \nabla w_{0} \vert }^{2}+{V(x_{0})}w _{0}^{2}\bigr)+ \biggl(\frac{1}{4}-\frac{1}{p+1}\biggr) \int _{B_{\rho _{1}}(0)}K(x_{0}) \vert w _{0} \vert ^{p+1}\le c_{0}-\frac{2}{3}d_{0}, $$

which contradicts (2.4). □

To obtain the multiplicity result of problem (2.1), we need the following two results:

Lemma 2.4

(see Theorem 5.20 of [28])

If \(I_{\varepsilon } \vert _{\mathcal{N}_{\varepsilon }}\) is bounded frow below and satisfies the \((PS)_{c}\) condition for any \(c\in [\inf_{\mathcal{N}_{\varepsilon }}I_{\varepsilon }, d]\), then \(I_{\varepsilon } \vert _{\mathcal{N}_{\varepsilon }}\) has a minimum and \(I_{\varepsilon }^{d}\) contains at least \(\mathrm{cat}_{I_{\varepsilon }^{d}}{I_{ \varepsilon }^{d}}\) critical points of \(I_{\varepsilon } \vert _{\mathcal{N}_{\varepsilon }}\), where \(I_{\varepsilon }^{d}:=\{u \in \mathcal{N}_{\varepsilon }: I_{\varepsilon }(u)\le d\}\).

Lemma 2.5

(see Lemma 4.3 of [3])

Let Γ, \(\varOmega ^{+}\), \(\varOmega ^{-}\) be closed sets with \(\varOmega ^{-}\subset \varOmega ^{+}\). Let \(\varPhi :\varOmega ^{-}\to \varGamma \), \(\beta :\varGamma \to \varOmega ^{+}\) be two continuous maps such that \(\beta \circ \varPhi \) is homotopically equivalent to the embedding \(\mathrm{Id}:\varOmega ^{-}\to \varOmega ^{+}\). Then \(\mathrm{cat}_{\varGamma }(\varGamma )\ge \mathrm{cat}_{\varOmega ^{+}}(\varOmega ^{-})\).

Palais–Smale condition

In this section, we prove that the functional \(I_{\varepsilon }\) satisfies the Palais–Smale condition on \(\mathcal{N}_{\varepsilon }\). We say that \(I_{\varepsilon }\vert _{\mathcal{N}_{\varepsilon }}\) satisfies the \((PS)_{c}\) condition if any sequence \(\{u_{n}\}\subset \mathcal{N} _{\varepsilon }\) such that \(I_{\varepsilon }(u_{n})\to c\) and \(\|I_{\varepsilon }'(u_{n})\|_{\ast }\to 0\) contains a convergent subsequence. Here \(\|I_{\varepsilon }'(u_{n})\|_{*}\) denotes the norm of the derivative of \(I_{\varepsilon }\) restricted to \(\mathcal{N} _{\varepsilon }\) at the point \(u_{n}\in \mathcal{N}_{\varepsilon }\).

Lemma 3.1

For \(\varepsilon >0\) sufficiently small, the constrained functional \(I_{\varepsilon }\vert _{\mathcal{N}_{\varepsilon }}\) satisfies the \((PS)_{c}\) condition for \(c< c_{\infty }\), where \(c_{\infty }\) is defined in Remark 1.3.

Proof

Since the ground energy \(c_{\varepsilon }\) of functional \(I_{\varepsilon }\) satisfies \(\limsup_{\varepsilon \to 0^{+}}c_{\varepsilon }\le c_{0}\) and \(c_{0}< c_{\infty }\) by (1.9), we know that the set \(\{u\in \mathcal{N}_{\varepsilon }:I_{\varepsilon }(u)< c_{\infty }\}\) is not empty for \(\varepsilon >0\) sufficiently small.

Let \(\{u_{n}\}\subset \mathcal{N}_{\varepsilon }\) be such that

$$ I_{\varepsilon }(u_{n})\to c\quad \mbox{and}\quad \bigl\Vert I_{\varepsilon }'(u_{n}) \bigr\Vert _{*} \to 0. $$
(3.1)

As \(\{u_{n}\}\subset \mathcal{N}_{\varepsilon }\), we have

$$\begin{aligned} I_{\varepsilon }(u_{n}) =&\biggl(\frac{1}{2}- \frac{1}{p+1}\biggr) \int _{\mathbb{R} ^{3}}a{ \vert \nabla u_{n} \vert }^{2}+{V(\varepsilon x)}u_{n}^{2}+ \biggl( \frac{1}{4}- \frac{1}{p+1}\biggr)b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u_{n} \vert }^{2}\biggr)^{2} \\ \ge &\biggl(\frac{1}{2}-\frac{1}{p+1}\biggr) \Vert u_{n} \Vert _{E_{\varepsilon }}^{2}. \end{aligned}$$

Then by \(I_{\varepsilon }(u_{n})\to c\) and \(c< c_{\infty }\) we know that \(\{u_{n}\}\) is bounded in \(E_{\varepsilon }\). Thus there exists \(u\in E_{\varepsilon }\) and if necessary a subsequence of \(\{u_{n}\}\) such that \(u_{n}\rightharpoonup u\) in \(E_{\varepsilon }\), \(u_{n}\to u\) in \(L_{\mathrm{loc}}^{\tau }(\mathbb{R}^{3})\) for \(1\le \tau <6\), and \(u_{n}\to u\) a.e. on \(\mathbb{R}^{3}\). We have to prove that \(u_{n}\to u\) strongly in \(E_{\varepsilon }\) and \(u\in \mathcal{N}_{\varepsilon }\).

First we show that if \(\|I_{\varepsilon }'(u_{n})\|_{*}\to 0\) then \(I_{\varepsilon }'(u_{n})\to 0\), which implies that \(\{u_{n}\}\) is a \((PS)_{c}\) sequence for the unconstrained functional \(I_{\varepsilon }\). Indeed, by \(\|I_{\varepsilon }'(u_{n})\|_{*}\to 0\), there exists \(\mu _{n}\in \mathbb{R}\) such that \(I_{\varepsilon }'(u_{n})-\mu _{n}S'(u _{n})\to 0\), where \(S(u)= \langle I_{\varepsilon }'(u),u \rangle \). Then we have

$$ 0=S(u_{n})= \bigl\langle I_{\varepsilon }'(u_{n}),u_{n} \bigr\rangle =\mu _{n} \bigl\langle S'(u _{n}),u_{n} \bigr\rangle +o(1). $$

From Lemma 2.2, there exists \(\tau _{\varepsilon }>0\) such that \(\langle S'(u_{n}),u_{n} \rangle \le -\tau _{\varepsilon }\), then by the above equality we have that \(\mu _{n}\to 0\) as \(n\to \infty \). By the definition of \(S(u)\) and the boundedness of \(\{u_{n}\}\) in \(E_{\varepsilon }\), we know that \(\|S'(u_{n})\|\) is bounded. Thus from \(I_{\varepsilon }'(u_{n})=\mu _{n}S'(u_{n})+o(1)\) we can get \(I_{\varepsilon }'(u_{n}) \to 0\), as \(n\to \infty \).

Now we prove if \(u_{n}\to u\) strongly in \(E_{\varepsilon }\), then \(u\in \mathcal{N}_{\varepsilon }\). Since \(u_{n}\in \mathcal{N}_{ \varepsilon }\), we have

$$ { \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla u_{n} \vert }^{2}+{V(\varepsilon x)}u_{n} ^{2}}\bigr)+ b \biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u_{n} \vert }^{2}\biggr)^{2}= \int _{\mathbb{R}^{3}}K(\varepsilon x) \vert u_{n} \vert ^{p+1}. $$

If \(u_{n}\to u\) in \(E_{\varepsilon }\), then passing to a limit in the above equality, we have

$$ { \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla u \vert }^{2}+{V( \varepsilon x)}u^{2}}\bigr)+ b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert }^{2} \biggr)^{2}= \int _{\mathbb{R}^{3}}K( \varepsilon x) \vert u \vert ^{p+1}, $$

which implies that \(u\in \mathcal{N}_{\varepsilon }\).

In order to prove \(u_{n}\to u\) in \(E_{\varepsilon }\), it suffices to show that, for any \(\delta >0\), there exists \(R>0\) such that

$$ { \int _{ \vert x \vert \ge R}\bigl(a{ \vert \nabla u_{n} \vert }^{2}+{V(\varepsilon x)}u_{n}^{2}}\bigr)< \delta \quad \mbox{for each }n\in \mathbb{N}^{+}. $$
(3.2)

Indeed, by (3.2), we first show that \(u_{n}\to u\) in \(L^{p+1}(\mathbb{R}^{3})\). For any \(\delta >0\), by (3.2), there exists \(R>0\) such that

$$ \biggl( \int _{ \vert x \vert \ge R} \vert u_{n} \vert ^{p+1} \biggr)^{\frac{1}{p+1}}\le C\biggl( \int _{ \vert x \vert \ge R}a { \vert \nabla u_{n} \vert }^{2}+{V(\varepsilon x)}u_{n}^{2} \biggr)^{\frac{1}{2}} \le C\delta ^{\frac{1}{2}}, $$
(3.3)

where \(C>0\) is a constant which is not dependent on R and n. Since \(u_{n}\to u\) in \(L_{\mathrm{loc}}^{p+1}(\mathbb{R}^{3})\), we have that for the fixed δ and R in (3.3), there exists \(N\in \mathbb{N}^{+}\) such that, for \(n>N\),

$$ \biggl( \int _{ \vert x \vert \le R} \vert u_{n}-u \vert ^{p+1} \biggr)^{\frac{1}{p+1}}\le \delta . $$
(3.4)

Combining (3.3) and (3.4), we can know that \(u_{n}\to u\) in \(L^{p+1}(\mathbb{R}^{3})\). Next we show that by (3.2), we can prove \(u_{n}\to u\) in \(E_{\varepsilon }\). Note that

$$\begin{aligned} &\bigl\langle I_{\varepsilon }'(u_{n})-I_{\varepsilon }'(u),u_{n}-u \bigr\rangle \\ &\quad =\biggl(a+b \int _{\mathbb{R}^{3}}{ \vert \nabla u_{n} \vert }^{2}\biggr) \int _{\mathbb{R}^{3}} \nabla u_{n}\cdot \nabla (u_{n}-u) \\ &\qquad {}+ \int _{\mathbb{R}^{3}}V(\varepsilon x) (u_{n}-u)^{2} - \biggl(a+b \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert }^{2}\biggr) \int _{\mathbb{R}^{3}}\nabla u \cdot \nabla (u_{n}-u)\\ &\qquad{}- \int _{\mathbb{R}^{3}}K(\varepsilon x) \bigl( \vert u_{n} \vert ^{p-1}u_{n}- \vert u \vert ^{p-1}u\bigr) (u _{n}-u) \\ &\quad =\biggl(a+b \int _{\mathbb{R}^{3}}{ \vert \nabla u_{n} \vert }^{2}\biggr)\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla (u_{n}-u) \bigr\vert ^{2}+ \int _{\mathbb{R}^{3}}V( \varepsilon x) (u_{n}-u)^{2} \\ &\qquad{}+b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u_{n} \vert } ^{2}- \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert }^{2}\biggr) \cdot \int _{\mathbb{R}^{3}}\nabla u\cdot \nabla (u_{n}-u)\\ &\qquad{}- \int _{\mathbb{R}^{3}}K(\varepsilon x) \bigl( \vert u_{n} \vert ^{p-1}u_{n}- \vert u \vert ^{p-1}u\bigr) (u _{n}-u)\\ &\quad \ge \Vert u_{n}-u \Vert _{\varepsilon }^{2}-b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert } ^{2}- \int _{\mathbb{R}^{3}}{ \vert \nabla u_{n} \vert }^{2}\biggr) \int _{\mathbb{R}^{3}} \nabla u\cdot \nabla (u_{n}-u)\\ &\qquad{}- \int _{\mathbb{R}^{3}}K(\varepsilon x) \bigl( \vert u_{n} \vert ^{p-1}u_{n}- \vert u \vert ^{p-1}u\bigr) (u _{n}-u), \end{aligned}$$

which implies that

$$\begin{aligned} \Vert u_{n}-u \Vert _{\varepsilon }^{2} \le& \bigl\langle I_{\varepsilon }'(u_{n})-I _{\varepsilon }'(u),u_{n}-u \bigr\rangle + b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert }^{2}- \int _{\mathbb{R}^{3}}{ \vert \nabla u_{n} \vert }^{2}\biggr)\\ &{}\cdot \int _{\mathbb{R}^{3}}\nabla u\cdot \nabla (u_{n}-u) + \int _{\mathbb{R}^{3}}K(\varepsilon x) \bigl( \vert u_{n} \vert ^{p-1}u_{n}- \vert u \vert ^{p-1}u\bigr) (u _{n}-u). \end{aligned}$$

Since \(u_{n}\rightharpoonup u\) and \(I_{\varepsilon }'(u_{n})\to 0\), we have \(\langle I_{\varepsilon }'(u_{n})-I_{\varepsilon }'(u),u_{n}-u \rangle \to 0\) as \(n\to \infty \). By the boundedness of \(\{u_{n}\}\) in \(E_{\varepsilon }\), we have

$$ b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert }^{2}- \int _{\mathbb{R}^{3}}{ \vert \nabla u_{n} \vert }^{2}\biggr) \int _{\mathbb{R}^{3}}\nabla u\cdot \nabla (u_{n}-u) \to 0, $$

as \(n\to \infty \). Furthermore,

$$\begin{aligned} & \biggl\vert \int _{\mathbb{R}^{3}}K(\varepsilon x) \bigl( \vert u_{n} \vert ^{p-1}u_{n}- \vert u \vert ^{p-1}u\bigr) (u _{n}-u) \biggr\vert \\ &\quad \le \Vert K \Vert _{\infty }\biggl( \int _{\mathbb{R}^{3}} \bigl\vert \vert u_{n} \vert ^{p-1}u_{n}- \vert u \vert ^{p-1}u \bigr\vert ^{ \frac{p+1}{p}}\biggr)^{\frac{p}{p+1}} \biggl( \int _{\mathbb{R}^{3}} \vert u_{n}-u \vert ^{p+1} \biggr)^{ \frac{1}{p+1}}. \end{aligned}$$

Since \(\{u_{n}\}\) is bounded in \(L^{p+1}(\mathbb{R}^{3})\) and \(u_{n}\to u\) in \(L^{p+1}(\mathbb{R}^{3})\), we have

$$ \int _{\mathbb{R}^{3}}K(\varepsilon x) \bigl( \vert u_{n} \vert ^{p-1}u_{n}- \vert u \vert ^{p-1}u\bigr) (u _{n}-u)\to 0, $$

as \(n\to \infty \). Thus we have \(\|u_{n}-u\|_{\varepsilon }\to 0\) as \(n\to \infty \), i.e., \(u_{n}\to u\) in \(E_{\varepsilon }\).

Now we are in a position to prove (3.2) to complete the proof of Lemma 3.1. By contradiction assume that for some subsequence \(\{u_{n_{k}}\}\) (we denote \(\{u_{k}\}\) for the simplicity of notations) and some \(\delta _{0}>0\)

$$ \int _{ \vert x \vert \ge k}a{ \vert \nabla u_{k} \vert }^{2}+{V(\varepsilon x)}u_{k}^{2} \ge \delta _{0} $$
(3.5)

for any k. By the choice of c and Remark 1.3, there exists \(\eta >0\) such that \(c< c(V_{\infty }-\eta , K_{\infty }+ \eta )=:c_{\eta }\) and \(c_{\eta }< c_{\infty }\). Let \(R(\eta )>0\) be an integer and such that \(V(\varepsilon x)\ge V_{\infty }-\eta \) and \(K(\varepsilon x)\le K_{\infty }+\eta \) for \(|x|\ge R(\eta )\). For any \(r>0\), we define \(A_{r}:=\{x\in \mathbb{R}^{3}:r\le |x|\le r+1\}\). Then as in [8], we can know that there exists \(r>R(\eta )\) and if necessary a subsequence of \(\{u_{k}\}\) such that

$$ \int _{A_{r}}a{ \vert \nabla u_{k} \vert }^{2}+{V(\varepsilon x)}u_{k}^{2}< \eta . $$
(3.6)

Now we fix \(r=r(\eta )>R(\eta )\) so that (3.6) holds. Let \(\rho \in C^{\infty }(\mathbb{R}^{3})\) be such that \(\rho (x)=0\) for \(|x|\le r\), \(\rho (x)=1\) for \(|x|\ge r+1\), \(0\le \rho \le 1\), and \(|\nabla \rho (x)|\le 2\) for any \(x\in \mathbb{R}^{3}\). Define \(w_{k}:=\rho u_{k}\).

As \(u_{k}\in \mathcal{N}_{\varepsilon }\), we have

$$ I_{\varepsilon }(u_{k})=\biggl(\frac{1}{2}-\frac{1}{p+1} \biggr) \int _{\mathbb{R} ^{3}}a{ \vert \nabla u_{k} \vert }^{2}+{V(\varepsilon x)}u_{k}^{2}+ \biggl( \frac{1}{4}- \frac{1}{p+1}\biggr)b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u_{k} \vert }^{2}\biggr)^{2}. $$

Define

$$ \bar{I}_{\varepsilon }(w_{k}):=\biggl(\frac{1}{2}- \frac{1}{p+1}\biggr) \int _{\mathbb{R}^{3}}a{ \vert \nabla w_{k} \vert }^{2}+{V(\varepsilon x)}w_{k} ^{2}+ \biggl( \frac{1}{4}-\frac{1}{p+1}\biggr)b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla w _{k} \vert }^{2}\biggr)^{2}, $$

then by the definition of \(w_{k}\) and (3.6), we have

$$ \bar{I}_{\varepsilon }(w_{k})\le I_{\varepsilon }(u_{k})+O( \eta ), $$
(3.7)

where \(|O(\eta )|< C\eta \) and \(C>0\) is a constant.

Now let \(\theta _{k}>0\) be such that \(\theta _{k}w_{k}\in \mathcal{N} _{\varepsilon }\). If \(\theta _{k}\le 1\) (up to a subsequence) for \(k=1,2,3,\ldots\) , then by (3.7) we have

$$\begin{aligned} &I_{\varepsilon }(\theta _{k}w_{k}) \\ &\quad =\biggl(\frac{1}{2}-\frac{1}{p+1}\biggr)\theta _{k}^{2} \int _{\mathbb{R}^{3}}a { \vert \nabla w_{k} \vert }^{2}+{V(\varepsilon x)}w_{k}^{2}+ \biggl( \frac{1}{4}- \frac{1}{p+1}\biggr)b\theta _{k}^{4} \biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla w_{k} \vert } ^{2}\biggr)^{2} \\ &\quad \le \biggl(\frac{1}{2}-\frac{1}{p+1}\biggr) \int _{\mathbb{R}^{3}}a{ \vert \nabla w_{k} \vert } ^{2}+{V(\varepsilon x)}w_{k}^{2}+ \biggl( \frac{1}{4}-\frac{1}{p+1}\biggr)b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla w_{k} \vert }^{2}\biggr)^{2} \\ &\quad =\bar{I}_{\varepsilon }(w_{k})\le I_{\varepsilon }(u_{k})+O(\eta ). \end{aligned}$$
(3.8)

Now we assume \(\theta _{k}>1\) for each k. Since \(\langle I_{\varepsilon }'(\theta _{k}w_{k}),\theta _{k}w_{k} \rangle =0\), we have \(\langle I_{ \varepsilon }'(w_{k}),w_{k} \rangle >0\) by Lemma 2.1. Denote \(\tilde{I}(w_{k})\) by

$$ \tilde{I}(w_{k})={ \int _{\mathbb{R}^{3}}a{ \vert \nabla w_{k} \vert }^{2}+{V( \varepsilon x)}w_{k}^{2}}+ b \int _{\mathbb{R}^{3}}{ \vert \nabla u_{k} \vert }^{2} \int _{\mathbb{R}^{3}}{ \vert \nabla w_{k} \vert }^{2}- \int _{\mathbb{R}^{3}}K( \varepsilon x) \vert w_{k} \vert ^{p+1}, $$

then we have

$$ \bigl\vert \bigl\langle I_{\varepsilon }'(u_{k}),w_{k} \bigr\rangle -\tilde{I}(w_{k}) \bigr\vert \le C_{1} \int _{A_{r}}a{ \vert \nabla u_{k} \vert }^{2}+{V(\varepsilon x)}u_{k} ^{2}, $$

where \(C_{1}>0\) is a constant which does not depend on r. Then, by (3.1) and (3.6), we have \(\tilde{I}(w_{k})=O( \eta )+o(1)\). Since \(\langle I_{\varepsilon }'(w_{k}),w_{k} \rangle >0\), we have

$$\begin{aligned} \begin{aligned}[b] \bigl\langle I_{\varepsilon }'(w_{k}),w_{k} \bigr\rangle & =\tilde{I}(w_{k})+ b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla w_{k} \vert }^{2}- \int _{\mathbb{R}^{3}}{ \vert \nabla u_{k} \vert }^{2}\biggr) \int _{\mathbb{R}^{3}}{ \vert \nabla w_{k} \vert }^{2} \\ & \le\tilde{I}(w_{k})+ b \int _{A_{r}}{ \vert \nabla w_{k} \vert }^{2} \int _{\mathbb{R} ^{3}}{ \vert \nabla w_{k} \vert }^{2}\\ & \le \tilde{I}(w_{k})+O(\eta )=O(\eta )+o(1). \end{aligned} \end{aligned}$$
(3.9)

By the definition of \(w_{k}\) and (3.5), we have

$$ \int _{\mathbb{R}^{3}}a{ \vert \nabla w_{k} \vert }^{2}+{V(\varepsilon x)}w_{k} ^{2}\ge \delta _{0}+O(\eta ). $$
(3.10)

Then by \(\theta _{k}w_{k}\in \mathcal{N}_{\varepsilon }\), (3.9) and (3.10), we have that \(\{\theta _{k}\}\) is bounded and (see the similar result (6.13) in [8])

$$ \theta _{k}=1+O(\eta )+o(1). $$
(3.11)

Thus by (3.7) and (3.11) we have

$$ I_{\varepsilon }(\theta _{k}w_{k})\le I_{\varepsilon }(u_{k})+O(\eta ). $$
(3.12)

From (3.8) and (3.12), up to a subsequence of \(\{w_{k}\}\), we have

$$ I_{\varepsilon }(\theta _{k}w_{k})\le I_{\varepsilon }(u_{k})+O(\eta ). $$
(3.13)

Let \(\tilde{w}_{k}:=\theta _{k}w_{k}\), and let \(\tilde{\theta }_{k}\) be such that \(\tilde{\theta }_{k}\tilde{w}_{k}\in \mathcal{N}_{\eta }\), the Nehari manifold defined as in Remark 1.3, with \(m=V_{\infty }-\eta \) and \(n=K_{\infty }+\eta \) in (1.7). From

$$\begin{aligned} & \int _{\mathbb{R}^{3}}a{ \vert \nabla \tilde{w}_{k} \vert }^{2}+(V_{\infty }- \eta )\tilde{w}_{k}^{2}+ b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla \tilde{w} _{k} \vert }^{2}\biggr)^{2} \\ &\quad \le \int _{\mathbb{R}^{3}}a{ \vert \nabla \tilde{w}_{k} \vert }^{2}+V(\varepsilon x)\tilde{w}_{k}^{2}+ b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla \tilde{w}_{k} \vert } ^{2}\biggr)^{2} \\ &\quad = \int _{\mathbb{R}^{3}}K(\varepsilon x) \vert \tilde{w}_{k} \vert ^{p+1} \\ &\quad \le \int _{\mathbb{R}^{3}}(K_{\infty }+\eta ) \vert \tilde{w}_{k} \vert ^{p+1}, \end{aligned}$$

we can know that \(\tilde{\theta }_{k}\le 1\), the above equality holds because \(\tilde{w}_{k}=\theta _{k}w_{k}\in \mathcal{N}_{\varepsilon }\). Now, by Lemma 2.1, the function

$$\begin{aligned} h(t) :=&\frac{t^{2}}{2}{ \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla \tilde{w}_{k} \vert } ^{2}+{V(\varepsilon x)}\tilde{w}_{k}^{2}} \bigr)+ \frac{t^{4}}{4}b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla \tilde{w}_{k} \vert }^{2}\biggr)^{2} \\ &{}-\frac{t^{p+1}}{p+1} \int _{\mathbb{R}^{3}}K(\varepsilon x) \vert \tilde{w} _{k} \vert ^{p+1} \end{aligned}$$

is nondecreasing for \(t\in (0, 1)\). Thus, by (3.13) and (3.1),

$$\begin{aligned} c_{\eta } \le& \frac{\tilde{\theta }_{k}^{2}}{2} \int _{\mathbb{R}^{3}}a { \vert \nabla \tilde{w}_{k} \vert }^{2}+(V_{\infty }-\eta )\tilde{w}_{k}^{2} + \frac{b}{4}\tilde{\theta }_{k}^{4}\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla \tilde{w}_{k} \vert }^{2}\biggr)^{2} \\ &{}-\frac{\tilde{\theta }_{k}^{p+1}}{p+1} \int _{\mathbb{R}^{3}}(K_{ \infty }+\eta ) \vert \tilde{w}_{k} \vert ^{p+1} \le \frac{\tilde{\theta }_{k} ^{2}}{2} \int _{\mathbb{R}^{3}}a{ \vert \nabla \tilde{w}_{k} \vert }^{2}+V(\varepsilon x)\tilde{w}_{k}^{2} \\ &{}+\frac{b}{4}\tilde{\theta }_{k}^{4}\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla \tilde{w}_{k} \vert }^{2}\biggr)^{2}- \frac{\tilde{\theta }_{k}^{p+1}}{p+1} \int _{\mathbb{R}^{3}}K(\varepsilon x) \vert \tilde{w}_{k} \vert ^{p+1} \\ \le&\frac{1}{2} \int _{\mathbb{R}^{3}}a{ \vert \nabla \tilde{w}_{k} \vert }^{2}+V( \varepsilon x)\tilde{w}_{k}^{2}+ \frac{b}{4}\biggl( \int _{\mathbb{R}^{3}} { \vert \nabla \tilde{w}_{k} \vert }^{2}\biggr)^{2}- \frac{1}{p+1} \int _{\mathbb{R}^{3}}K( \varepsilon x) \vert \tilde{w}_{k} \vert ^{p+1} \\ =&I_{\varepsilon }(\tilde{w}_{k})=I_{\varepsilon }(\theta _{k}w_{k}) \le I_{\varepsilon }(u_{k})+O(\eta ) \le c+O(\eta )+o(1). \end{aligned}$$

Letting \(k\to \infty \), \(\eta \to 0\) and by the continuity of \(c_{\eta }\) with respect to η (see Remark 1.3), we know that \(c_{\infty }\le c\), a contradiction which concludes the proof. □

The maps \(\varPhi _{\varepsilon }\) and \(\beta _{\varepsilon }\)

In this section we construct two mappings \(\varPhi _{\varepsilon }\) and \(\beta _{\varepsilon }\) in order to apply Lemma 2.5 to prove Theorem 1.1.

Let \(\delta >0\) be fixed and \(\eta \in C_{0}^{\infty }(\mathbb{R}^{3})\) such that \(0\le \eta \le 1\), \(\eta =1\) on \(B_{1}(0)\), \(\eta =0\) on \(\mathbb{R}^{3}\setminus B_{2}(0)\), \(|\nabla \eta |\le C\) for some \(C>0\). For any \(y\in M\) (defined in (1.5)), we define

$$ \varPsi _{\varepsilon ,y}(x)=\eta \biggl(\frac{\varepsilon x-y}{\sqrt{\varepsilon }}\biggr)w^{y} \biggl(\frac{\varepsilon x-y}{\varepsilon }\biggr), $$

where \(w^{y}\) is the unique positive ground state solution (see [23]) of

$$ \textstyle\begin{cases} -(a+b{\int _{\mathbb{R}^{3}}}{ \vert \nabla v \vert }^{2})\Delta v+V(y)v=K(y) \vert v \vert ^{p-1}v \quad \mbox{in } \mathbb{R}^{3}, \\ v>0, \quad v\in H^{1}(\mathbb{R}^{3}). \end{cases} $$
(4.1)

Let w be such that \(w^{y}=\lambda w(\mu x)\), where \(\mu ^{2}=V(y)\) and \(\lambda =(V(y)/K(y))^{\frac{1}{p-1}}\), then w satisfies

$$ \textstyle\begin{cases} -(a+b\frac{\lambda ^{2}}{\mu }{\int _{\mathbb{R}^{3}}}{ \vert \nabla v \vert }^{2}) \Delta v+v= \vert v \vert ^{p-1}v \quad \mbox{in } \mathbb{R}^{3}, \\ v>0, \quad v\in H^{1}(\mathbb{R}^{3}). \end{cases} $$
(4.2)

Since \(\frac{\lambda ^{2}}{\mu }=\frac{V^{\frac{2}{p-1}-\frac{1}{2}}(y)}{K ^{\frac{2}{p-1}}(y)}\), then by the definition of M we know that, for any \(y\in M\),

$$ \frac{\lambda ^{2}}{\mu }=\frac{V^{\frac{2}{p-1}-\frac{1}{2}}(y)}{K ^{\frac{2}{p-1}}(y)}\equiv L, $$

where L is a positive constant. Thus we have that, for \(y\in M\),

$$ w^{y}=\lambda w(\mu x), $$
(4.3)

where w is the unique positive ground state solution of (4.2) with \(\frac{\lambda ^{2}}{\mu }=L\).

Now let \(t_{\varepsilon ,y}>0\) be such that \(I_{\varepsilon }(t_{ \varepsilon ,y}\varPsi _{\varepsilon ,y})=\max_{t\ge 0}I_{\varepsilon }(t \varPsi _{\varepsilon ,y})\) and \({\frac{dI_{\varepsilon }(t\varPsi _{\varepsilon ,y})}{dt}}|_{t=t_{\varepsilon ,y}}=0\). We define \(\varPhi _{\varepsilon }:M\to \mathcal{N}_{\varepsilon }\) by \(\varPhi _{\varepsilon }(y):=t_{\varepsilon ,y}\varPsi _{\varepsilon ,y}\).

Lemma 4.1

Uniformly for \(y\in M\), we have

$$ \lim_{\varepsilon \to 0^{+}}I_{\varepsilon }\bigl(\varPhi _{\varepsilon }(y)\bigr)=c _{0}, $$
(4.4)

where \(c_{0}\) is defined in Remark 1.3.

Proof

We first show that \(t_{\varepsilon ,y}\to 1\) as \(\varepsilon \to 0^{+}\). Since \(t_{\varepsilon ,y}\varPsi _{\varepsilon ,y}=\varPhi _{\varepsilon }(y) \in \mathcal{N}_{\varepsilon }\), we have

$$ { \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla \varPsi _{\varepsilon ,y} \vert }^{2}+{V( \varepsilon x)}\varPsi _{\varepsilon ,y}^{2}}\bigr)+ t_{\varepsilon ,y}^{2}b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla \varPsi _{\varepsilon ,y} \vert }^{2}\biggr)^{2}= t_{ \varepsilon ,y}^{p-1} \int _{\mathbb{R}^{3}}K(\varepsilon x) \vert \varPsi _{\varepsilon ,y} \vert ^{p+1}. $$

By the definition of \(\varPsi _{\varepsilon ,y}\) and (4.3), after a change of variable, we get

$$\begin{aligned} &{ \int _{B_{\frac{\mu }{\sqrt{\varepsilon }}}(0)}\biggl(La{ \vert \nabla w \vert } ^{2}+\frac{L}{\mu ^{2}}{V\biggl(\frac{\varepsilon x}{\mu }+y \biggr)}w^{2}}\biggr)+ o(1)+ t_{\varepsilon ,y}^{2}b\biggl(L \int _{B_{\frac{\mu }{\sqrt{\varepsilon }}}(0)} { \vert \nabla w \vert }^{2}+ o(1)\biggr)^{2} \\ &\quad =t_{\varepsilon ,y}^{p-1}\biggl( \frac{L}{K(y)} \int _{B_{\frac{\mu }{\sqrt{\varepsilon }}}(0)}K\biggl(\frac{\varepsilon x}{ \mu }+y\biggr)w^{p+1}+o(1) \biggr). \end{aligned}$$
(4.5)

By the definition of μ and (H), we know that as \(\varepsilon \to 0^{+}\), \(\frac{\mu }{\sqrt{\varepsilon }}\to + \infty \) uniformly for \(y\in M\). Moreover, for \(|x|\le \frac{\mu }{\sqrt{ \varepsilon }}\), \(|\frac{\varepsilon x}{\mu }+y|\) is bounded and \(\frac{\varepsilon x}{\mu }+y\to y\) as \(\varepsilon \to 0^{+}\) uniformly for \(y\in M\). Then we have as \(\varepsilon \to 0^{+}\) and uniformly for \(y\in M\),

$$\begin{aligned}& \int _{B_{\frac{\mu }{\sqrt{\varepsilon }}}(0)}La{ \vert \nabla w \vert }^{2} \to La \int _{\mathbb{R}^{3}}{ \vert \nabla w \vert }^{2}, \\& \int _{B_{\frac{\mu }{\sqrt{\varepsilon }}}(0)}\frac{L}{\mu ^{2}} {V\biggl(\frac{\varepsilon x}{\mu }+y \biggr)}w^{2}\to \int _{\mathbb{R}^{3}}\frac{L}{ \mu ^{2}}{V(y)}w^{2}=L \int _{\mathbb{R}^{3}}w^{2}, \\& \frac{L}{K(y)} \int _{B_{\frac{\mu }{\sqrt{\varepsilon }}}(0)}K\biggl(\frac{ \varepsilon x}{\mu }+y\biggr)w^{p+1} \to \frac{L}{K(y)} \int _{\mathbb{R}^{3}}K(y)w ^{p+1}=L \int _{\mathbb{R}^{3}}w^{p+1}. \end{aligned}$$

Now assume that there exist \(t_{0}\), \(T_{0}\) such that \(0< t_{0}\le t _{\varepsilon ,y}\le T_{0}\), and let \(t_{\varepsilon ,y}\to T>0\) as \(\varepsilon \to 0^{+}\), then by the above estimates we have

$$ \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla w \vert }^{2}+w^{2}\bigr)+bT^{2}L\biggl( \int _{\mathbb{R}^{3}} \vert \nabla w \vert ^{2} \biggr)^{2}= T^{p-1} \int _{\mathbb{R}^{3}}w ^{p+1}. $$

Since w is the ground state solution of (4.2), we have

$$ \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla w \vert }^{2}+w^{2}\bigr)+bL\biggl( \int _{\mathbb{R} ^{3}} \vert \nabla w \vert ^{2} \biggr)^{2}= \int _{\mathbb{R}^{3}}w^{p+1}, $$

these imply that

$$ \bigl(T^{p-1}-1\bigr) \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla w \vert }^{2}+w^{2}\bigr)+ bL\bigl(T^{p-1}-T ^{2} \bigr) \biggl( \int _{\mathbb{R}^{3}} \vert \nabla w \vert ^{2} \biggr)^{2}=0. $$

If \(T<1\), then the left part of the above equality is less than 0, and if \(T>1\), it will be larger than 0, which yields that \(T=1\).

Now we prove that \(t_{\varepsilon ,y}\nrightarrow 0\). Otherwise, from (4.5), we have

$$ \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla w \vert }^{2}+w^{2}\bigr)=0, $$

which is a contradiction. Also from (4.5), we have that \(t_{\varepsilon ,y}\nrightarrow +\infty \) as \(p-1>2\). By the above arguments, we can see that \(t_{\varepsilon ,y}\to 1\) uniformly for \(y\in M\) as \(\varepsilon \to 0^{+}\).

Note that

$$\begin{aligned} I_{\varepsilon }\bigl(\varPhi _{\varepsilon }(y) \bigr) =&I_{\varepsilon }(t_{\varepsilon ,y}\varPsi _{\varepsilon ,y}) \\ =&\frac{t_{\varepsilon ,y}^{2}}{2}\biggl(L \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla w \vert }^{2}+w^{2}\bigr)+o(1)\biggr)+ \frac{t_{\varepsilon ,y}^{4}}{4}b\biggl( \biggl( \int _{\mathbb{R}^{3}}L \vert \nabla w \vert ^{2} \biggr)^{2}+o(1)\biggr) \\ &{}-\frac{t_{\varepsilon ,y}^{p+1}}{p+1}\biggl(L \int _{\mathbb{R}^{3}}w^{p+1}+o(1)\biggr) =\frac{L}{2} \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla w \vert }^{2}+w^{2}\bigr)+\frac{L ^{2}}{4}b\biggl( \int _{\mathbb{R}^{3}} \vert \nabla w \vert ^{2} \biggr)^{2} \\ &{}-\frac{L}{p+1} \int _{\mathbb{R}^{3}}w^{p+1}+o(1)=LI_{L}(w)+o(1), \end{aligned}$$
(4.6)

where \(I_{L}\) is the energy functional of equation (4.2) with \(\frac{\lambda ^{2}}{\mu }=L\) in it. Let \(I^{y}\) be the energy functional of (4.1), then we have

$$\begin{aligned} c_{0} =&I^{y}\bigl(w^{y}\bigr) \\ =&\frac{1}{2} \int _{\mathbb{R}^{3}}a{ \bigl\vert \nabla w^{y} \bigr\vert }^{2}+V(y) \bigl(w^{y}\bigr)^{2}+ \frac{b}{4}\biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{y} \bigr\vert ^{2}\biggr)^{2} - \frac{1}{p+1} \int _{\mathbb{R}^{3}}K(y) \bigl(w^{y}\bigr)^{p+1} \\ =&\frac{1}{2}\frac{\lambda ^{2}}{\mu } \int _{\mathbb{R}^{3}}a{ \vert \nabla w \vert }^{2}+ \frac{1}{2}\frac{\lambda ^{2}}{\mu ^{3}} \int _{\mathbb{R}^{3}}V(y)w ^{2}+ \frac{b}{4} \frac{\lambda ^{4}}{\mu ^{2}}\biggl( \int _{\mathbb{R}^{3}} \vert \nabla w \vert ^{2} \biggr)^{2} \\ &{}-\frac{1}{p+1}\frac{\lambda ^{p+1}}{\mu ^{3}} \int _{\mathbb{R}^{3}}K(y)w ^{p+1} \\ =&L\biggl(\frac{1}{2} \int _{\mathbb{R}^{3}}\bigl(a{ \vert \nabla w \vert }^{2}+w^{2}\bigr)+ \frac{L}{4}b\biggl( \int _{\mathbb{R}^{3}} \vert \nabla w \vert ^{2} \biggr)^{2} -\frac{1}{p+1} \int _{\mathbb{R}^{3}}w^{p+1}\biggr)=LI_{L}(w), \end{aligned}$$

thus from (4.6), we prove that \(\lim_{\varepsilon \to 0^{+}}I_{\varepsilon }(\varPhi _{\varepsilon }(y))=c _{0}\). □

Remark 4.2

If there is no competing potential function \(K(x)\) in (1.1), i.e., \(K(x)\equiv 1\), then in equation (4.1), \(K(y)\equiv 1\). In this case, for different \(y\in M\), \(V(y)\) is the same, then the positive ground state solution \(w^{y}\) of (4.1) is the same function for every \(y\in M\). But in our case, because of the competing function \(K(y)\) in (4.1), the ground state solution \(w^{y}\) may change for different \(y\in M\), this causes troubles in the proof of (4.4), and we develop the technique of rescaling to solve the problem.

Let \(\rho >0\) be such that \(M_{\delta }\subset B_{\rho }(0):=\{x \in \mathbb{R}^{3}:|x|\le \rho \}\). Define \(\gamma :\mathbb{R}^{3} \to \mathbb{R}^{3}\) by \(\gamma (x)=x\) for \(|x|\le \rho \) and \(\gamma (x)=\rho x/|x|\) for \(|x|\ge \rho \). Consider the mapping \(\beta _{\varepsilon }:\mathcal{N}_{\varepsilon }\to \mathbb{R}^{3}\) given by \(\beta _{\varepsilon }(u):=\frac{\int _{\mathbb{R}^{3}}\gamma (\varepsilon x)u^{2}}{\int _{\mathbb{R}^{3}}u^{2}}\), then as the proof in [8] and by (4.3) we have that

$$ \lim_{\varepsilon \to 0^{+}}\beta _{\varepsilon }\bigl(\varPhi _{\varepsilon }(y)\bigr)=y\quad \mbox{uniformly for }y \in M. $$
(4.7)

Now define \(h(\varepsilon ):=\sup_{y\in M}|I_{\varepsilon }( \varPhi _{\varepsilon }(y))-c_{0}|\), then Lemma 4.1 yields that \(h(\varepsilon )\to 0\) as \(\varepsilon \to 0^{+}\). Let

$$ \tilde{\mathcal{N}}_{\varepsilon }:=\bigl\{ u\in \mathcal{N}_{\varepsilon }:I _{\varepsilon }(u)\le c_{0}+h(\varepsilon )\bigr\} , $$
(4.8)

then by the definition of \(h(\varepsilon )\) we know that, for any \(y\in M\) and \(\varepsilon >0\), \(\varPhi _{\varepsilon }(y)\in \tilde{\mathcal{N}}_{\varepsilon }\) and \(\tilde{\mathcal{N}}_{\varepsilon }\neq \emptyset \).

Lemma 4.3

Let \(\varepsilon _{n}\to 0^{+}\) and \(u_{n}\in \tilde{\mathcal{N}}_{ \varepsilon _{n}}\). Then there exists \(\{y_{n}\}\subset \mathbb{R}^{3}\) such that the sequence \(\{u_{n}(x+y_{n})\}\) has a convergent subsequence in \(H^{1}(\mathbb{R}^{3})\) and \(\varepsilon _{n}y_{n}\to y\in M\).

Proof

As in [23], for \(u_{n}\in \tilde{\mathcal{N}}_{\varepsilon _{n}}\), we define a measure \(\mu _{n}\) on \(\mathbb{R}^{3}\) by

$$ \mu _{n}(\varOmega )= \int _{\varOmega }\biggl[\frac{1}{4}\bigl(a{ \vert \nabla u_{n} \vert }^{2}+ {V(\varepsilon _{n} x)}u_{n}^{2}\bigr)+ \biggl(\frac{1}{4}- \frac{1}{p+1}\biggr)K( \varepsilon _{n} x) \vert u_{n} \vert ^{p+1}\biggr]. $$

Since \(0\le \mu _{n}(\mathbb{R}^{3})=I_{\varepsilon _{n}}(u_{n})\le c _{0}+h(\varepsilon _{n})\), then along a subsequence if necessary, as \(\varepsilon _{n}\to 0^{+}\),

$$ \mu _{n}\bigl(\mathbb{R}^{3}\bigr)\to \tilde{c}\le c_{0}. $$
(4.9)

Moreover, let \(\bar{V}=\inf_{x\in \mathbb{R}^{3}}V(x)\) and \(\bar{K}= \sup_{x\in \mathbb{R}^{3}}K(x)\), then by Lemma 3.3 of [23], \(\tilde{c}\ge \bar{c}>0\) where \(\bar{c}=c(\bar{V}, \bar{K})\) is defined in Remark 1.3.

By the concentration-compactness lemma of P.L. Lions in [19] and as the proof in Lemma 4.1 in [23], we know that there exists a sequence \(\{y_{n}\}\subset \mathbb{R}^{3}\) such that, for any \(h>0\), there is \(\rho >0\) such that

$$ \int _{B_{\rho }(y_{n})}d\mu _{n}\ge \tilde{c}-h. $$
(4.10)

Now we prove that \(\{\varepsilon _{n}y_{n}\}\) is bounded. Otherwise, assume that \(|\varepsilon _{n}y_{n}|\to \infty \) as \(n\to \infty \). Since \(\mu _{n}(\mathbb{R}^{3})\) is bounded, we know that \(w_{n}:=u_{n}(x+y _{n})\) is bounded in \(H^{1}(\mathbb{R}^{3})\). Therefore there exists \(w_{0}\in H^{1}(\mathbb{R}^{3})\) such that up to a subsequence, \(w_{n}\rightharpoonup w_{0}\) in \(H^{1}(\mathbb{R}^{3})\), \(w_{n}\to w _{0}\) in \(L_{\mathrm{loc}}^{\tau }(\mathbb{R}^{3})\) for \(1\le \tau <6\), and almost everywhere in \(\mathbb{R}^{3}\). Furthermore, by (4.10), we can prove that \(w_{n}\to w_{0}\) in \(L^{\tau }(\mathbb{R}^{3})\) for \(1\le \tau <6\) and \(w_{0}\neq 0\) in \(H^{1}(\mathbb{R}^{3})\). Let \(\theta _{\infty }>0\) be such that \(\theta _{\infty }w_{0}\in \mathcal{N}_{\infty }\), the Nehari manifold associated to (1.7) with \(m=V_{\infty }\) and \(n=K_{\infty }\) in it. Then as the proof in the Appendix of [8], we have \(\theta _{\infty } \le 1\); and furthermore, \(\{\varepsilon _{n}y_{n}\}\) is bounded.

Assume that \(\{\varepsilon _{n}y_{n}\}\) converges to some y (up to a subsequence), we now prove that \(y\in M\) and that \(w_{n}\to w\) strongly in \(H^{1}(\mathbb{R}^{3})\). Since \(u_{n}\in \mathcal{N}_{\varepsilon _{n}}\) and \(w_{n}=u_{n}(x+y_{n})\), we have

$$ { \int _{\mathbb{R}^{3}}a{ \vert \nabla w_{n} \vert }^{2}+{V(\varepsilon _{n} x+ \varepsilon _{n}y_{n})}w_{n}^{2}}+ b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla w _{n} \vert }^{2}\biggr)^{2}= \int _{\mathbb{R}^{3}}K(\varepsilon _{n} x+\varepsilon _{n}y_{n}) \vert w_{n} \vert ^{p+1}. $$

Taking the lower limit of both sides of the above equality and by \(\varepsilon _{n}y_{n}\to y\), we have

$$ { \int _{\mathbb{R}^{3}}a{ \vert \nabla w \vert }^{2}+{V(y)}w^{2}}+ b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla w \vert }^{2} \biggr)^{2}\le \int _{\mathbb{R}^{3}}K(y) \vert w \vert ^{p+1}. $$

Now let \(\theta _{y}>0\) be such that \(\theta _{y} w\in \mathcal{N}_{y}\), the Nehari manifold associated to (4.1), we have that \(\theta _{y}\le 1\) by Lemma 2.1. Let \(I^{y}\) be the energy functional associated to (4.1). Then

$$\begin{aligned} c_{0} \le& c\bigl(V(y),K(y)\bigr)\le I^{y}( \theta _{y} w) \\ =&\frac{1}{4}\theta _{y}^{2} \int _{\mathbb{R}^{3}}a{ \vert \nabla w \vert }^{2}+ {V(y)}w^{2}+ \biggl(\frac{1}{4}-\frac{1}{p+1}\biggr)\theta _{y}^{p+1} \int _{\mathbb{R}^{3}}K(y) \vert w \vert ^{p+1} \\ \le &\frac{1}{4} \int _{\mathbb{R}^{3}}a{ \vert \nabla w \vert }^{2}+{V(y)}w^{2}+ \biggl( \frac{1}{4}-\frac{1}{p+1}\biggr) \int _{\mathbb{R}^{3}}K(y) \vert w \vert ^{p+1} \\ \le& \liminf_{n\to \infty }\biggl[{\frac{1}{4} \int _{\mathbb{R}^{3}}a{ \vert \nabla w_{n} \vert }^{2}+{V(\varepsilon _{n} x+\varepsilon _{n}y_{n})}w_{n} ^{2}}+ \biggl(\frac{1}{4}-\frac{1}{p+1}\biggr) \\ &{}\cdot \int _{\mathbb{R}^{3}}K(\varepsilon _{n} x+\varepsilon _{n}y_{n}) \vert w _{n} \vert ^{p+1}\biggr]=\liminf_{n\to \infty }I_{\varepsilon _{n}}(u_{n})= \tilde{c}\le c_{0}, \end{aligned}$$
(4.11)

which implies that \(\theta _{y}=1\) and \(c(V(y),K(y))=c_{0}\). Thus we have \(y\in M\). Moreover, \(I_{y}(w)=c_{0}\), hence w is a ground state solution of (4.1). The strong convergence \(w_{n}\to w\) in \(L^{\tau }(\mathbb{R}^{3})\) for \(1\le \tau <6\) and (4.11) give

$$ \liminf_{n\to \infty }{ \int _{\mathbb{R}^{3}}a{ \vert \nabla w_{n} \vert }^{2}+ {V(\varepsilon _{n} x+\varepsilon _{n}y_{n})}w_{n}^{2}}= \int _{\mathbb{R}^{3}}a{ \vert \nabla w \vert }^{2}+{V(y)}w^{2}. $$
(4.12)

From (4.12) we can prove that \(w_{n}\to w\) in \(H^{1}(\mathbb{R}^{3})\). □

Lemma 4.4

For any \(\delta >0\), we have

$$ \lim_{\varepsilon \to 0^{+}}\sup_{u\in \tilde{N}_{\varepsilon }}\operatorname{dist}\bigl( \beta _{\varepsilon }(u),M_{\delta }\bigr)=0. $$

Proof

The proof is similar to the proof of Lemma 5.1 in [8] or Lemma 4.7 in [13], we omit it here. □

Proof of Theorem 1.1

For \(\delta >0\), by Lemma 4.1, Lemma 4.4, and (4.7), there exists \(\varepsilon _{\delta }>0\) such that, for any \(\varepsilon \in (0, \varepsilon _{\delta })\), the diagram

$$ M \xrightarrow{\varPhi _{\varepsilon }}\tilde{\mathcal{N}}_{\varepsilon } \xrightarrow{\beta _{\varepsilon }}M_{\delta } $$

is well defined. Moreover, by (4.7), the mapping \(\beta _{\varepsilon }\circ \varPhi _{\varepsilon }\) is homotopic to the inclusion \(\mathrm{Id}:M\to M_{\delta }\). Now set \(\tilde{\mathcal{N}}_{\varepsilon }^{+}:=\tilde{\mathcal{N}}_{\varepsilon }\cap \{u\in \mathcal{N}_{ \varepsilon }:u\ge 0\mbox{ in }\mathbb{R}^{3}\}\), then similar to [8] (or [7]), by Lemma 2.5 we have that \(\mathrm{cat}_{\tilde{\mathcal{N}}_{\varepsilon }}( \tilde{\mathcal{N}}_{\varepsilon }^{+})\ge \mathrm{cat}_{M_{\delta }}(M)\); and furthermore, \(\mathrm{cat}_{\tilde{\mathcal{N}}_{\varepsilon }}( \tilde{\mathcal{N}}_{\varepsilon })\ge 2\mathrm{cat}_{M_{\delta }}(M)\). Lemma 2.4 shows that \(I_{\varepsilon }\) has at least \(2 \mathrm{cat}_{M_{\delta }}(M)\) critical points on \(\tilde{\mathcal{N}}_{\varepsilon }\). Now, in order to prove Theorem 1.1, we only need to show that the critical point \(u\in \tilde{\mathcal{N}}_{\varepsilon }\) cannot change sign for sufficiently small \(\varepsilon >0\). Indeed, if \(u=u^{+} +u^{-}\) with \(u^{+} \not \equiv 0\) and \(u^{-} \not \equiv 0\). First, because \(u\in \tilde{\mathcal{N}}_{\varepsilon }\), we have

$$ I_{\varepsilon }(u)\le c_{0}+h(\varepsilon ), $$
(5.1)

where \(h(\varepsilon )\to 0\) as \(\varepsilon \to 0^{+}\). Since \(u\in \mathcal{N}_{\varepsilon }\), we have

$$\begin{aligned} I_{\varepsilon }(u) =&\biggl(\frac{1}{2}- \frac{1}{p+1}\biggr) \Vert u \Vert _{E_{\varepsilon }}^{2}+ \biggl(\frac{1}{4}-\frac{1}{p+1}\biggr)b\biggl( \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert } ^{2} \biggr)^{2} \\ \ge &\biggl(\frac{1}{2}-\frac{1}{p+1}\biggr) \bigl\Vert u^{+} \bigr\Vert _{E_{\varepsilon }}^{2}+ \biggl( \frac{1}{4}-\frac{1}{p+1}\biggr)b\biggl( \int _{\mathbb{R}^{3}}{ \bigl\vert \nabla u^{+} \bigr\vert }^{2}\biggr)^{2} \\ &{}+\biggl(\frac{1}{2}-\frac{1}{p+1}\biggr) \bigl\Vert u^{-} \bigr\Vert _{E_{\varepsilon }}^{2}+ \biggl( \frac{1}{4}-\frac{1}{p+1}\biggr)b\biggl( \int _{\mathbb{R}^{3}}{ \bigl\vert \nabla u^{-} \bigr\vert }^{2}\biggr)^{2} \\ =:&\tilde{I}_{\varepsilon }\bigl(u^{+}\bigr)+\tilde{I}_{\varepsilon } \bigl(u^{-}\bigr), \end{aligned}$$
(5.2)

where \(\tilde{I}_{\varepsilon }(u)\) is defined by \(\tilde{I}_{\varepsilon }(u):=(\frac{1}{2}-\frac{1}{p+1})\|u\|_{E_{\varepsilon }}^{2}+ ( \frac{1}{4}-\frac{1}{p+1})b(\int _{\mathbb{R}^{3}}{|\nabla u|}^{2})^{2}\).

Since \(u^{+} \not \equiv 0\), there exists \(t^{+}>0\) such that \(t^{+}u^{+}\in \mathcal{N}_{\varepsilon }\). Multiplying equation (2.1) by \(u^{+}\) and integrating over \(\mathbb{R}^{3}\), we have

$$ \biggl(a+b \int _{\mathbb{R}^{3}}{ \vert \nabla u \vert }^{2}\biggr) \int _{\mathbb{R}^{3}}{ \bigl\vert \nabla u^{+} \bigr\vert }^{2}+ { \int _{\mathbb{R}^{3}}{V(\varepsilon x)} {u^{+}} ^{2}}- \int _{\mathbb{R}^{3}}K(\varepsilon x) \bigl\vert u^{+} \bigr\vert ^{p+1}=0, $$

which implies that

$$\begin{aligned} \bigl\langle I_{\varepsilon }' \bigl(u^{+}\bigr),u^{+} \bigr\rangle =&\biggl(a+b \int _{\mathbb{R}^{3}}{ \bigl\vert \nabla u^{+} \bigr\vert }^{2}\biggr) \int _{\mathbb{R}^{3}}{ \bigl\vert \nabla u^{+} \bigr\vert }^{2}+ { \int _{\mathbb{R}^{3}}{V(\varepsilon x)} {u^{+}} ^{2}} \\ &{}- \int _{\mathbb{R}^{3}}K(\varepsilon x) \bigl\vert u^{+} \bigr\vert ^{p+1}< 0. \end{aligned}$$
(5.3)

Since \(t^{+}u^{+}\in \mathcal{N}_{\varepsilon }\), we have \(\langle {I _{\varepsilon }}'(t^{+}u^{+}),t^{+}u^{+} \rangle =0\). Then from (5.3) we get that \(0< t^{+}<1\). Now

$$\begin{aligned} c_{\varepsilon } =&\inf_{u\in \mathcal{N}_{\varepsilon }}I_{\varepsilon }(u) \le {I_{\varepsilon }}\bigl(t^{+}u^{+}\bigr) \\ =&\biggl(\frac{1}{2}-\frac{1}{p+1}\biggr){t^{+}}^{2} \bigl\Vert u^{+} \bigr\Vert _{E_{\varepsilon }} ^{2}+ \biggl(\frac{1}{4}-\frac{1}{p+1}\biggr)b{t^{+}}^{4} \biggl( \int _{\mathbb{R}^{3}} { \bigl\vert \nabla u^{+} \bigr\vert }^{2}\biggr)^{2} \\ < &\biggl(\frac{1}{2}-\frac{1}{p+1}\biggr) \bigl\Vert u^{+} \bigr\Vert _{E_{\varepsilon }}^{2} +\biggl( \frac{1}{4}-\frac{1}{p+1}\biggr)b\biggl( \int _{\mathbb{R}^{3}}{ \bigl\vert \nabla u^{+} \bigr\vert }^{2}\biggr)^{2}= \tilde{I}_{\varepsilon } \bigl(u^{+}\bigr). \end{aligned}$$
(5.4)

Similar to (5.4), we can also prove that \(\tilde{I}_{\varepsilon }(u^{-})>c_{\varepsilon }\). Now by (5.2), we have that \(I_{\varepsilon }(u)\ge \tilde{I}_{\varepsilon }(u^{+})+\tilde{I}_{ \varepsilon }(u^{-})>2c_{\varepsilon }\), which contradicts (5.1) by Lemma 2.3. Thus we can assume that there exist at least \(\mathrm{cat}_{M_{\delta }}(M)\) critical points that are positive on \(\mathbb{R}^{3}\) and by the maximum principle they are strictly positive. Now the proof of Theorem 1.1 is complete.

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Acknowledgements

This work was completed while the author was visiting the Academy of Mathematics and Systems Science at the Chinese Academy of Sciences, and the author is grateful for the kind hospitality of the host institution. The author would like to express sincere thanks to the anonymous referees for carefully reading the manuscript and valuable comments and suggestions.

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The paper is supported by the Youth Foundation of Qilu Normal University (2017L0602), Natural Science Foundation of Shandong Province (ZR2018MA009).

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Sun, D. Multiple positive solutions to Kirchhoff equations with competing potential functions in \(\mathbb{R}^{3}\). Bound Value Probl 2019, 85 (2019). https://doi.org/10.1186/s13661-019-1198-9

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MSC

  • 35J60
  • 35J20
  • 35B38

Keywords

  • Kirchhoff equation
  • Multiplicity
  • Competing potential functions
  • Nonlocal problems