In present paper, we concern with the following semi-linear Kirchhoff type equation

$\begin{equation}\label{1.1} \left\{ \begin{array}{ll} -(\varepsilon^2a+\varepsilon b \int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+V(x) u =f(u),\,\, &x\in \mathbb{R}^3, \\ u\in H^1(\mathbb{R}^3), \end{array} \right. \end{equation}$

(1.1)

where $\varepsilon>0$ is a parameter, $a,b>0$ are positive constants, $V(x)$ is a Hölder continuous function satisfying

$(\bf V_1)$ $V(x)\geq \alpha>0,x\in \mathbb{R}^3$ for some constant $\alpha>0.$

$(\bf V_2)$ There exists a bounded domain $\Lambda $ compactly contained in $\mathbb{R}^3$ such that

$V_0:=\inf\limits_{x\in \Lambda } V(x)< \inf\limits_{x\in \partial \Lambda } V(x)$

and $f\in C^1(\mathbb{R})$ satisfying

$(\bf f_1)$ $f(s)=o(s^3)$ as $s\rightarrow 0$.

$(\bf f_2)$ $\displaystyle\lim\limits_{|s|\rightarrow +\infty}\frac{|f(s)|}{|s|^p}=0$ for some $3<p<5$.

$(\bf f_3)$ there exists some $\theta>4$ such that

$0<\theta F(s)\leq sf(s), \forall s\neq 0,$

where $F(s)=\int_0^sf(t)dt$.

$(\bf f_4)$ $\frac{f(s)}{|s|^3}$ is increasing for any $s\neq 0$.

Equation (1.1) with $a=1,b=0$ and $\mathbb{R}^3$ replaced by $\mathbb{R}^N$, reduces to the well-known Schrödinger equation

$\begin{equation}\label{1.2} -\varepsilon^2\Delta u+V(x)u=f(u),\ \ x\in \mathbb{R}^N. \end{equation}$

(1.2)

Equation (1.2) arises in different models. For instance, they are involved with the existence of standing waves of the nonlinear Schrödinger equations

$\begin{equation}\label{1.3} i\varepsilon\partial_t z=-\varepsilon^2\Delta z+(V(x)+E)z-f(z),\ \ \forall x\in \mathbb{R}^N, \end{equation}$

(1.3)

where $f(s)=|s|^{p-2}s,2<p<2^*:=2N/(N-4)$. A standing waves of (1.3) is a solution of the form $z(x,t)=\exp(-iEt/\varepsilon)u(x)$, where $u$ is a solution of (1.2).

For (1.1), if $\varepsilon=1,V(x)=0$ and $\mathbb{R}^3$ replaced by $\Omega$, it reduces to the following Kirchhoff type problem

$\begin{equation}\label{1.4} \left\{ \begin{array}{ll} -(a+ b \int_{\Omega}|\nabla u|^2dx)\Delta u=f(x,u),\,\, &x\in \Omega, \\ u=0, \,\, &x\in \partial\Omega. \end{array} \right. \end{equation}$

(1.4)

where $\Omega\subset \mathbb{R}^3$ is a smooth bounded domain.

Equation (1.4) is related to the stationary analogue of the equation

$\begin{equation}\label{1.5} \left\{ \begin{array}{ll} u_{tt}-(a+ b\displaystyle\int_{\Omega}|\nabla u|^2dx)\Delta u=f(x,u),\,\, &x\in \Omega, \\ u=0, \,\, &x\in \partial\Omega. \end{array} \right. \end{equation}$

(1.5)

Equations of this type were first proposed by Kirchhoff in [19] to describe the transversal oscillations of a stretched string. Equation (1.5) began to attract much attention since the work of Lions [18] introduced an abstract framework to the problem. More results can refer to, for example [16-17]. Meanwhile, the presence of the term $(\int_{\Omega}|\nabla u|^2dx)\Delta u$ implies that the above two equation are no longer a point-wise identity. This phenomenon provokes some mathematical difficulties, which make the study of such a class of problem particularly interesting.

In recent years, a lot of work has been done by many authors related to equation (1.2) and (1.4). We can refer to [1-3, 5-7, 9-12, 14] and their references therein.

More recently, He and Zou in [15] considered the following general equation

$\begin{equation}\label{1.6} \left\{ \begin{array}{ll} -(\varepsilon^2a+\varepsilon b \int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+V(x) u =f(u),\,u>0,\, &x\in \mathbb{R}^3, \\ u\in H^1(\mathbb{R}^3).\ \end{array} \right. \end{equation}$

(1.6)

By using Ljusternik-Schnirelmnn theory (see [13]) and mini-max methods, the multiplicity of positive solutions, which concentrate on the minima of $V(x)$ as $\varepsilon\rightarrow 0$, are obtained. But to the best of our knowledge, the existence and concentration behavior of the sign changing solutions to (1.1) have not ever been studied. Moreover, as far as we know, the existence and concentration behavior of node solutions are very interesting in both mathematicians and physicians. Fortunately, in [4], the author studied the following equation

$\begin{equation}\label{1.7} \left\{ \begin{array}{ll} -\varepsilon^2\Delta u+V(x)u=f(u),\,\, &x\in \Omega, \\ u=0, \,\, &x\in \partial\Omega. \end{array} \right. \end{equation}$

(1.7)

Under some given conditions with $V(x), f(u)$, the existence of node solutions was obtained and such a solution has just one positive and negative peaks which are located around local minimal of $V(x)$.

Motivated by the above papers, we study the existence and concentration behavior of nodal solutions of problem (1.1). In our present paper, we mainly employ the method used in [4]. However, compared with [4], the term $(\int_{\Omega}|\nabla u|^2dx)\Delta u$ and the lack of compactness of the embed-ding of $H^1(\mathbb{R}^3)\hookrightarrow L^p(\mathbb{R}^3), 2<p<6$ cause us more difficulties. So we need to find some new arguments and our work is meaningful.

Our main result is as follows:

Theorem 1.1 Suppose that $V(x)$ satisfies $(\bf V_1)-(V_2)$ and $f$ satisfies $(\bf f_1)-(f_4)$, then there exists $\varepsilon_0>0$ such that problem (1.1) possesses a nodal solutions $u_{\varepsilon}\in H^1(\mathbb{R}^3)$ for every $\varepsilon\in (0,\varepsilon_0)$. Moreover, $u_\varepsilon$ possesses just one positive maximum point $P_{\varepsilon}^1\in \Lambda $ and one negative minimum point $P_{\varepsilon}^2\in \Lambda $. We also obtain $\displaystyle\lim\limits_{\varepsilon \rightarrow 0}V(P_{\varepsilon}^i)=V_0 (i=1,2)$ and

$|u_\varepsilon(x)|\leq M\Big[\exp(-\beta|\frac{x-P_{\varepsilon}^1}{\varepsilon}|)+\exp(-\beta|\frac{x-P_{\varepsilon}^2}{\varepsilon}|)\Bigr], $

where $M,\beta$ are some positive constants.

To verify Theorem 1.1, we mainly employ the framework used in [4]. We first exploit the truncation method to modify the nonlinearity $f(u)$ in order to obtain the existence of a nodal solution. Furthermore, to show the phenomenon of concentration, we establish an upper estimate of the energy for the solution and make a careful study of its profile obtaining a relation between peak points, which imply that these points are concentrated around local minimal of $V$.

In this section, we introduce some notations and prove the existence of a nodal solution to equation (1.1). Throughout this paper, we denote by $H$ the Hilbert space given by

$ H=\{u\in H^1(\mathbb{R}^3):\int_{\mathbb{R}^3}V(x)u^2<+\infty\}$

endowed with the norm denote by $\|u\|=(\int_{\mathbb{R}^3}(|\nabla u|^2+u^2)dx )^{1/2}$. It is clear that solutions of (1.1) are the critical points of the functional $I_\varepsilon:H\rightarrow \mathbb{R}$ given by

$I_\varepsilon(u)=\frac12 \int_{\mathbb{R}^3}(\varepsilon^2 a|\nabla u|^2+V(x)u^2)dx+\frac b4(\int_{\mathbb{R}^3}|\nabla u|^2dx)^2-\int_{\mathbb{R}^3}F(u)dx,$

where $F(u)=\displaystyle\int_0^uf(s)ds.$ By $\bf(f_1)-(f_2)$, $I_\varepsilon$ is well defined and $I_\varepsilon\in C^1(H,\mathbb{R}).$ Similar to the argument in [2], we choose $k>0$ such that $k>\frac{\theta}{\theta-2}>1$. Take $a_1>0, a_2<0$ such that $\frac{f(a_i)}{a_i}=\frac{\alpha}{k}, i=1,2,$ where $\alpha$ is as in $\bf(V_1)$. Meanwhile, we set

$\begin{equation*} \tilde{f}(s)=\left\{\begin{array}{ll} \frac{\alpha s}{k},\,\, s>a_1 \ {\rm or}\ s<a_2, \\ f(s),\,\,a_2\leq s\leq a_1 \end{array} \right. \end{equation*}$

and define the functional

$g(x,s)=\chi_{\Lambda }(x)f(s)+(1-\chi_{\Lambda }(x))\tilde{f}(s),$

where $\chi_{\Lambda }(x)$ denotes the characteristic function of $\Lambda $. Using the conditions $\bf(f_1)-(f_4)$, it is easy to verify $g(x,s)$ is a Carethéodory function and satisfies the following conditions:

$(\bf g_1)$ $g(x,s)=o(|s|^3)$ as $s\rightarrow 0, $ uniformly in $x\in \mathbb{R}^3$.

$(\bf g_2)$ There exist some $3<p<5$ such that $\displaystyle\lim\limits_{|s|\rightarrow +\infty}\frac{g(x,s)}{|s|^p}=0$.

$(\bf g_3)$ There exist some $\theta>4$ such that

$\begin{eqnarray*} &&\ 0<\theta G(x,s)\leq sg(x,s), \forall x\in \Lambda , s\neq 0,\\ &&\ 0<2G(x,s)\leq sg(x,s)\leq \frac1k V(x)s^2, \forall x\in \mathbb{R}^3\setminus \Lambda , s\neq 0,\end{eqnarray*}$

where $G(x,s)=\displaystyle\int_0^sg(x,t)dt$.

$(\bf g_4)$ The function $\frac{g(x,s)}{|s|^3}$ is increasing for any $x\in \mathbb{R}^3, s\neq 0$.

In the following discussion, we consider the following penalized problem

$\begin{equation}\label{2.1} \left\{ \begin{array}{ll} -(\varepsilon^2a+ \varepsilon b \int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+V(x) u =g(x,u),\,\, &x\in \mathbb{R}^3, \\ u\in H. \end{array} \right. \end{equation}$

(2.1)

Note that if $u$ is a nodal solution of (2.1) with $a_2\leq u(x)\leq a_1$, then $u(x)$ is indeed a nodal solution to equation (1.1).

For equation (2.1), the corresponding energy functional $J_\varepsilon:H\rightarrow \mathbb{R}$ is defined by

$ J_\varepsilon (u)= \frac12 \int_{\mathbb{R}^3}(\varepsilon^2 a|\nabla u|^2+V(x)u^2)dx+\frac {\varepsilon b}4(\int_{\mathbb{R}^3}|\nabla u|^2dx)^2-\int_{\mathbb{R}^3}G(x,u)dx$

and for any $\varphi\in H,$

$ \langle J'_\varepsilon(u),\varphi\rangle= (\varepsilon^2 a +\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2 dx)\int_{\mathbb{R}^3}\nabla u\nabla \varphi dx+\int_{\mathbb{R}^3}V(x)u\varphi dx-\int_{\mathbb{R}^3}g(x,u)\varphi dx. $

To prove the existence of a nodal solution, we define

$ M_\varepsilon=\{u\in H:u^{\pm}\not\equiv 0, J_\varepsilon'(u)u^{\pm}=0\}$

and

$ c_\varepsilon=\inf\limits_{u\in M_\varepsilon}J_\varepsilon(u).$

Lemma 2.1  $c_\varepsilon$ is achieved by some $u_\varepsilon\in M_\varepsilon$. Moreover, $u_\varepsilon$ is a nodal solution of equation (2.1).

Proof Since $(\bf g_1)-(g_3)$, there exist constants $C>0, \mu>0$ such that for any $u\in M_\varepsilon$, we have

$ J_\varepsilon(u)\geq C\|u\|^2,\ \ \ \ \int_{\Lambda }|u^\pm|^{p+1}dx\geq \mu>0.$

Take a sequence $\{u_n\}\subset M_\varepsilon$ such that $J_\varepsilon(u_n)\rightarrow c_\varepsilon$, then $\{u_n\}$ is bounded in $H$. Thus, there exist a subsequence, still denoted by $\{u_n\}$ and a function $u\in H$ such that $u_n\rightharpoonup u$ in $H$. Thus, we have

$ \int_{\Lambda }|u^\pm|^{p+1}dx=\lim\limits_{n\rightarrow +\infty}\int_{\Lambda }|u_n^\pm|^{p+1}dx\geq \mu >0. $

Therefore, $u^\pm\not\equiv 0$.

Now, we claim that $J'_\varepsilon(u^\pm)u^\pm\leq 0$.

In fact, by the lower semi-continuous of the norm and Fatou Lemma, we derive that

$\begin{eqnarray*} &&\int_{\mathbb{R}^3}(\varepsilon^2 a |\nabla u^\pm|^2+V(x)|u^\pm|^2)dx +b\varepsilon(\int_{\mathbb{R}^3}|\nabla u^\pm|^2dx)^2 -\int_{\mathbb{R}^3\setminus {\Lambda }}g(x,u^\pm)u^\pm dx\\ &=& \int_{\mathbb{R}^3}\varepsilon^2 a |\nabla u^\pm|^2+\int_{\Lambda }V(x)|u^\pm|^2dx+\int_{\mathbb{R}^3\setminus {\Lambda }}(1-\frac1k)V(x)|u^\pm|^2dx+b\varepsilon(\int_{\mathbb{R}^3}|\nabla u^\pm|^2dx)^2\\ &&+\int_{\mathbb{R}^3\setminus {\Lambda }}(\frac1k V(x)|u^\pm|^2-g(x,u^\pm)u^\pm)dx \\ &\leq& \liminf\limits_{n\rightarrow \infty } \Bigr[\int_{\mathbb{R}^3}\varepsilon^2 a |\nabla u_n^\pm|^2dx+\int_{\Lambda }V(x)|u_n^\pm|^2dx+\int_{\mathbb{R}^3\setminus {\Lambda }}(1-\frac1k)V(x)|u_n^\pm|^2dx\\ &&+b\varepsilon(\int_{\mathbb{R}^3}|\nabla u_n^\pm|^2dx)^2\Bigr]+\liminf\limits_{n\rightarrow \infty } \Bigr[\int_{\mathbb{R}^3\setminus {\Lambda }}(\frac1k V(x)|u_n^\pm|^2-g(x,u_n^\pm)u_n^\pm )dx\Bigr]\\ &\leq& \liminf\limits_{n\rightarrow \infty }\Bigr[(\int_{\mathbb{R}^3}(\varepsilon^2 a |\nabla u_n^\pm|^2+\int_{\Lambda }V(x)|u_n^\pm|^2)dx+\int_{\mathbb{R}^3\setminus {\Lambda }}(1-\frac1k)V(x)|u_n^\pm|^2dx\\ &&+b\varepsilon(\int_{\mathbb{R}^3}|\nabla u_n^\pm|^2dx)^2)+\int_{\mathbb{R}^3\setminus {\Lambda }}(\frac1k V(x)|u_n^\pm|^2-g(x,u_n^\pm)u_n^\pm )dx\Bigr]\\ &=& \liminf\limits_{n\rightarrow \infty } \Bigr[\int_{\mathbb{R}^3}(\varepsilon^2 a |\nabla u_n^\pm|^2+V(x)|u_n^\pm|^2)dx +b\varepsilon(\int_{\mathbb{R}^3}|\nabla u_n^\pm|^2dx)^2 -\int_{\mathbb{R}^3\setminus {\Lambda }}g(x,u_n^\pm)u_n^\pm dx \Bigr]\\ &=& \liminf\limits_{n\rightarrow \infty } \Bigr[\int_{\mathbb{R}^3}g(x,u_n^\pm)u_n^\pm dx -\int_{\mathbb{R}^3\setminus {\Lambda }}g(x,u^\pm)u^\pm dx \Bigr]\\ &=& \liminf\limits_{n\rightarrow \infty } \int_{\Lambda }g(x,u_n^\pm)u_n^\pm dx\\ &=&\int_{\Lambda }g(x,u^\pm)u^\pm dx. \end{eqnarray*}$

Thus

$\int_{\mathbb{R}^3}(\varepsilon^2 a |\nabla u^\pm|^2+V(x)|u^\pm|^2)dx +b\varepsilon(\int_{\mathbb{R}^3}|\nabla u^\pm|^2dx)^2 -\int_{\mathbb{R}^3}g(x,u^\pm)u^\pm dx\leq 0, $

i.e., $\langle J'_\varepsilon (u^\pm),u^\pm\rangle\leq 0.$ By the above statement, there exists constant $t^\pm\in (0,1]$ such that $\langle J'_\varepsilon(t^\pm u^\pm),t^\pm u^\pm\rangle=0$, i.e., $u_\varepsilon=t^+u^++t^-u^-\in M_\varepsilon$. In addition, by $(\bf g_4),$ $(\bf V_1),$ $k>1$ and Fatou lemma, then we have

$\begin{eqnarray*} && J_\varepsilon(t^\pm u^\pm)=\frac12\int_{\mathbb{R}^3}(\varepsilon^2 a |t^\pm \nabla u^\pm|^2+V(x)|t^\pm u^\pm|^2)dx +\frac{\varepsilon b}{4}(\int_{\mathbb{R}^3}| t^\pm \nabla u^\pm|^2dx)^2-\int_{\mathbb{R}^3}G(x,t^\pm u^\pm) dx\\ &&= \int_{\mathbb{R}^3}[\frac14g(x,t^\pm u^\pm)t^\pm u^\pm-G(x,t^\pm u^\pm)]dx+\frac14\int_{\mathbb{R}^3}(\varepsilon^2 a | t^\pm \nabla u^\pm|^2+V(x)|t^\pm u^\pm|^2)dx\\ &&\leq \liminf\limits_{n\rightarrow \infty}\Bigr[ \int_{\mathbb{R}^3}(\frac14g(x,t^\pm u_n^\pm)t^\pm u_n^\pm-G(x,t^\pm u_n^\pm))dx+\frac14\int_{\mathbb{R}^3}(\varepsilon^2 a |t^\pm \nabla u_n^\pm|^2+V(x)|t^\pm u_n^\pm|^2)dx)\Bigr]\\ &&\leq \liminf\limits_{n\rightarrow \infty}\Bigr[ \int_{\mathbb{R}^3}(\frac14g(x,u_n^\pm)u_n^\pm-G(x,u_n^\pm))dx+\frac14\int_{\mathbb{R}^3}(\varepsilon^2 a |\nabla u_n^\pm|^2+V(x)| u_n^\pm|^2)dx\Bigr]\\ &&= \liminf\limits_{n\rightarrow \infty}J_\varepsilon(u_n^\pm). \end{eqnarray*}$

Thus

$\begin{eqnarray*} c_\varepsilon &\leq& J_\varepsilon(u_\varepsilon)=J_\varepsilon(t^+u^+)+J_\varepsilon(t^-u^-)\\ &\leq & \liminf\limits_{n\rightarrow \infty}[J_\varepsilon(u_n^+)+J_\varepsilon(u_n^-)]\\ &=& \liminf\limits_{n\rightarrow \infty}J_\varepsilon(u_n)\\ &=&c_\varepsilon, \end{eqnarray*}$

i.e., $c_\varepsilon$ is attained by the function $u_\varepsilon$. Furthermore, by the elliptic regularity arguments, $u_\varepsilon$ is a classical nodal solution of problem (2.1).

In this section, we turn to estimate the energy of $u_\varepsilon$. Let

$\begin{equation*} f_{\pm}(s)=\left\{\begin{array}{ll} f(s),\,\, \pm s\geq 0 \\ 0,\,\,{\rm otherwise}. \end{array} \right. \end{equation*}$

Suppose $\omega_\pm\in H^1(\mathbb{R}^3)$ are respectively the least energy nodal solutions of the following limit equation:

$ -(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+V_0u=f_{\pm}(u),\ \ x\in \mathbb{R}^3, $

that is $\omega_\pm$ satisfy $c_{V_0}^{\pm}:= J_{V_0}^{\pm}(\omega_\pm)=\inf\limits_{u\in H\setminus \{0\}}\sup\limits_{\tau\geq0}J_{V_0}^{\pm}(\tau u),$ where

$ J_{V_0}^{\pm}(\omega_\pm)=\frac12\int_{\mathbb{R}^3}(a|\nabla \omega_\pm|^2+V_0^\pm{\omega_\pm}^2)dx+\frac{b}{4}(\int_{\mathbb{R}^3}|\nabla \omega_\pm|^2dx)^2-\int_{\mathbb{R}^3}F_{\pm}(u)dx $

and $F_{\pm}(s)=\displaystyle\int_0^sf_{\pm}(t)dt.$

Without loss of generality, we assume $\omega_+(0)=\max\limits_{x\in \mathbb{R}^3}\omega_+(x)$, $\omega_-(0)=\min\limits_{x\in \mathbb{R}^3}\omega_-(x)$.

Lemma 3.1 Given $\varepsilon>0$, the function $u_\varepsilon$ satisfies

$ \limsup\limits_{\varepsilon\rightarrow 0}\varepsilon^{-3}J_\varepsilon(u_\varepsilon)\leq c_{V_0}^++c_{V_0}^-. $

Proof Let $x_0\in {\rm int}(\Lambda )$ be such that $V(x_0)=V_0$. Choosing $r>0$ such that $B_r(x_0)\subset {\rm int}(\Lambda )$ and $\eta$ is a smooth function, $0\leq \eta \leq 1$, $|\nabla \eta|\leq C$ and

$\begin{equation*} \eta(x)=\left\{\begin{array}{ll} 1,\,\, |x|\leq r/4, \\ 0,\,\,|x|\geq r/2. \end{array} \right. \end{equation*}$

Denote $\omega_{\varepsilon,\pm}(x)=\eta(x-x_0)\omega_{\pm}(\frac{x-x_0}{\varepsilon})$. By condition $(\bf g_1)-(\bf g_3)$, there exists $t_{\varepsilon, \pm}>0$ such that $J_{\varepsilon}(t_{\varepsilon,\pm}\omega_{\varepsilon,\pm})=\max\limits_{t>0}J_{\varepsilon}(t\omega_{\varepsilon,\pm})$, then

$ J_{\varepsilon}'(t_{\varepsilon,\pm}\omega_{\varepsilon,\pm})t_{\varepsilon,\pm}\omega_{\varepsilon,\pm}=0. $

Consider the function

$ \omega_{\varepsilon}=t_{\varepsilon,+}\omega_{\varepsilon,+}+t_{\varepsilon,-}\omega_{\varepsilon,-}, $

then $\omega^{\pm}=t_{\varepsilon,\pm}\omega_{\varepsilon,\pm}$, $J_\varepsilon'(\omega_{\varepsilon}^{\pm})\omega_{\varepsilon}^{\pm}=0$, i.e., $\omega_\varepsilon\in M_\varepsilon$, thus

$ c_\varepsilon\leq J_\varepsilon(\omega_\varepsilon)=J_\varepsilon(\omega_\varepsilon^+)+J_\varepsilon(\omega_\varepsilon^-). $

On the other hand, by direct computation we conclude that

$ J_\varepsilon(\omega_\varepsilon^\pm)=\varepsilon^3(c_{V_0}^\pm+o(1)), $

where $o(1)\rightarrow 0$ as $\varepsilon\rightarrow 0$. Thus we obtain our conclusion.

In this section, we make a careful analysis the profile of $u_\varepsilon$.

Lemma 4.1 The positive local maximum and negative local minimum points of $u_\varepsilon$ are both in $\Lambda $.

Proof Let $x_\varepsilon$ be a positive local maximum of $u_\varepsilon$. Suppose by contradiction that $x_\varepsilon\in \Lambda ^c$. Since $\Delta u_\varepsilon(x_\varepsilon)\leq 0$, using the definition of $g$, we have

$\begin{eqnarray*} \alpha u_\varepsilon(x_\varepsilon) &\leq& V(x_\varepsilon)u_\varepsilon(x_\varepsilon)\\ &\leq & -(\varepsilon^2a +\varepsilon b \int_{\mathbb{R}^3}|\nabla u_\varepsilon|^2dx)\Delta u_\varepsilon(x_\varepsilon)+V(x_\varepsilon)u_\varepsilon(x)\\ &=& \tilde{f}(u_\varepsilon(x_\varepsilon))\leq \frac{\alpha}{k}u_\varepsilon(x_\varepsilon). \end{eqnarray*}$

But the above estimate is impossible by the fact $k>\frac{\theta}{\theta-2}>1.$ A similar argument implies that every negative local minimum of $u_\varepsilon$ is also in $\Lambda $.

Lemma 4.2 Let $P_\varepsilon^1$ be a local maximum of $u_\varepsilon^+$ and $P_\varepsilon^2$ a local minimum of $u_\varepsilon^-,$ then

$(1)$ $u_\varepsilon(P_\varepsilon^1)\geq a_1$, $u_\varepsilon(P_\varepsilon^2)\leq a_2$,

$(2)$ $|\frac{P_\varepsilon^1-P_\varepsilon^2}{\varepsilon}|\rightarrow +\infty$ as $\varepsilon\rightarrow 0$.

Proof By Lemma 4.1, then $P_\varepsilon^1$, $P_\varepsilon^2\in \Lambda .$ Moreover, from the definition of $g$, then for $i=1,2$, we have

$\begin{eqnarray*} (\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u_\varepsilon|^2dx)\Delta u_\varepsilon(P_\varepsilon^i) &=& V(P_\varepsilon^i)u_\varepsilon(P_\varepsilon^i)-f(u_\varepsilon(P_\varepsilon^i))\\ &=& (V(P_\varepsilon^i)-\frac{f(u_\varepsilon(P_\varepsilon^i))}{u_\varepsilon(P_\varepsilon^i)})u_\varepsilon(P_\varepsilon^i). \end{eqnarray*}$

Meanwhile, as $\Delta u_\varepsilon(P_\varepsilon^1)\leq 0, \Delta u_\varepsilon(P_\varepsilon^2)\geq0, u_\varepsilon(P_\varepsilon^1)>0$ and $u_\varepsilon(P_\varepsilon^2)<0$, it follows that

$ V(P_\varepsilon^i)-\frac{f(u_\varepsilon(P_\varepsilon^i))}{u_\varepsilon(P_\varepsilon^i)}\leq 0, \ \ i=1,2, $

which together with $(V_1)$ imply

$\frac{f(u_\varepsilon(P_\varepsilon^i))}{u_\varepsilon(P_\varepsilon^i)}\geq \alpha>0,\ \ i=1,2.$

Consequence, if $a_2<u_\varepsilon(P_\varepsilon^i)<a_1$, $i=1,2$, then

$ \frac{\alpha}{k}=\frac{\frac{\alpha}{k}u_\varepsilon(P_\varepsilon^i)}{u_\varepsilon(P_\varepsilon^i)}\geq\frac{f(u_\varepsilon(P_\varepsilon^i))}{u_\varepsilon(P_\varepsilon^i)}\geq \alpha>0, $

which is a contradiction, thus $u_\varepsilon(P_\varepsilon^1)\geq a_1, u_\varepsilon(P_\varepsilon^2)\leq a_2$. Item (1) is proved. The proof of item 2 is similar to Lemma 3.2 in [4], here we omit it.

Lemma 4.3 If $\varepsilon_n\downarrow 0$ and $x_n^i\in \bar{\Lambda }, i=1,2$ are such that

$ u_{\varepsilon_n}(x_n^1)\geq b>0,\ \ \ u_{\varepsilon_n}(x_n^2)\leq -b<0, $

then

$ \lim\limits_{n\rightarrow \infty}V(x_n^i)=V_0\ (i=1,2). $

Proof The proof is similar to Proposition 4.1 in [4].

Lemma 4.4 If $m_{\varepsilon}^+=\max\limits_{x\in\partial \Lambda }u_{\varepsilon}^+(x),\ m_{\varepsilon}^-=\min\limits_{x\in\partial \Lambda }u_{\varepsilon}^-(x)$, then

$\lim\limits_{\varepsilon\rightarrow 0}m_{\varepsilon}^{\pm}=0. $

Moreover, for every $\varepsilon>0$ small enough, $u_\varepsilon$ possesses at most one positive local maximum $P_\varepsilon^1\in \Lambda $ and one negative local minimum $P_\varepsilon^2\in \Lambda $. Meanwhile, we have

$ \lim\limits_{\varepsilon\rightarrow 0}V(P_\varepsilon^i)=V_0=\min\limits_{x\in \bar{\Lambda }}V(x), i=1,2. $

Proof The proof is similar to Corollary 4.1 in [4].

To conclude the proof of Theorem 1.1, we only need to show that $a_2<u_{\varepsilon}(x)<a_1,\ \forall x\in \Lambda ^{c}$. By Lemma4.4, there exists $\varepsilon_0>0$ such that for all $0<\varepsilon<\varepsilon_0$, we have

$ |u_{\varepsilon}(x)|\leq A=\min\limits\{a_1, -a_2\}, \ \forall x\in\partial \Lambda . $

Using the same arguments in [2], the above inequality follows for $u_{\varepsilon}$ and $x\in \Lambda ^c$. Moreover, the arguments explored by [14], we can prove the following estimate

$|u_\varepsilon(x)|\leq M\Big[\exp(-\beta|\frac{x-P_{\varepsilon}^1}{\varepsilon}|)+\exp(-\beta|\frac{x-P_{\varepsilon}^2}{\varepsilon}|)\Bigr]$

for some constant $\beta>0$. So we complete the proof of Theorem 1.1.