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Existence and concentration of a nonlinear biharmonic equation with sign-changing potentials and indefinite nonlinearity

Abstract

We consider the following nonlinear biharmonic equations:

$$ \Delta^{2} u-\Delta u+ V_{\lambda }(x)u=f(x,u),\quad \text{in } \mathbb{R}^{N}, $$

where \(V_{\lambda }(x)\) is allowed to be sign-changing and f is an indefinite function. Under some suitable assumptions, the existence of nontrivial solutions and the high energy solutions are obtained by using variational methods. Moreover, the phenomenon of concentration of solutions is explored. The results extend the main conclusions in recent literature.

Introduction and main results

This paper concerns the existence results and the phenomenon of concentration of solutions for the following biharmonic equation:

$$ \Delta^{2} u-\Delta u+ V_{\lambda }(x)u=f(x,u),\quad \text{in } \mathbb{R}^{N}, $$
(1.1)

where \(\Delta^{2}=\Delta (\Delta )\) is the biharmonic operator, f is an indefinite function and the potential \(V_{\lambda }(x)=\lambda V ^{+}(x)-V^{-}(x)\) with \(V^{+}\) having a bounded potential well Ω whose depth is controlled by λ and \(V^{-}(x)\geq 0\) for all \(x\in \mathbb{R}^{N}\). Such an equation may arise in many fields of physics, such as describing the traveling waves in suspension bridge [17] and describing the static deflection of an elastic plate in fluid [7]. For more physical background of problem (1.1), we refer the readers to [11] and the references therein.

In the last two decades, the existence of bound states, ground states, semi-classical states (where \(\Delta^{2}\) is replaced by \(\varepsilon ^{4}\Delta^{2}\) for \(\varepsilon >0\) small), and infinitely many nontrivial solutions of biharmonic equations have been widely discussed under various conditions no matter on a bounded domain or on the whole space. Here we just give some references which are close to the problem we consider in this paper. For instance, Yin and Wu [25] studied problem (1.1) with various sets of assumptions on the nonlinearity \(f(x,u)\) (superquadraticity, subcriticality, etc.) and under the following conditions imposed on the potential \(V(x)\):

\((V_{1}')\) :

\(V\in C^{1}(\mathbb{R}^{N},\mathbb{R})\), \(\inf_{x\in \mathbb{R}^{N}}V(x)\geq a>0\), where a is a constant;

\((V_{2}')\) :

For each \(b>0\), \(\operatorname{meas}\{x\in \mathbb{R}^{N}:V(x) \leq b\}<\infty \), where meas denotes the Lebesgue measure in \(\mathbb{R}^{N}\).

They obtained the existence and infinitely many nontrivial solutions via variational methods. Soon after, Ye and Tang [23] improved these results. Here, we emphasize that conditions \((V_{1}')\) and \((V_{2}')\) are usually assumed to guarantee the compact embedding of the working space [33]. However, if \((V_{2}')\) is replaced by the following more general condition \((V_{2}'')\), the compactness of the embedding fails and this situation becomes more delicate.

\((V_{2}'')\) :

There exists \(b>0\) such that the set \(\{x\in \mathbb{R} ^{N}:V(x)\leq b\}\) is nonempty and has finite measure.

As far as we observe, there are few papers concerning this case. We mention that the authors in [16] investigated the existence and multiplicity results of problem (1.1) when conditions \((V_{1}')\) and \((V_{2}'')\) hold and the nonlinearity \(f(x,u)\) is superlinear at infinity and subcritical. Motivated by [16], Ye and Tang [24] studied problem (1.1) under the following more general case imposed on the potential \(V(x)\):

\((V_{1}'')\) :

\(V(x)\geq 0\) for all \(x\in \mathbb{R}^{N}\);

\((V_{2}^{\ast })\) :

There exists \(b>0\) such that the set \(\{x\in \mathbb{R}^{N}:V(x)\leq b\}\) has finite measure.

Under conditions \((V_{1}'')\) and \((V_{2}^{\ast })\) and more generic superlinear condition upon \(f(x,u)\), the authors obtained some results which unify and significantly improve the results in [16]. Moreover, by applying a new version of the symmetric mountain pass lemma, they also investigated infinitely many small-energy solutions of problem (1.1) when the nonlinearity \(f(x,u)\) is sublinear with mild assumptions different from [16]. For other interesting results on biharmonic equations, we refer readers to [4, 6, 8, 1315, 19, 21, 22, 2630, 32] and the references therein.

However, for most of these papers, the potential \(V(x)\) is always assumed to be positive. To the authors’ knowledge, there seems to be no result on the existence of solutions to problem (1.1) with sign-changing potential \(V_{\lambda }\). Indeed, this is an interesting question, and we mainly consider the following two problems in the present paper:

  1. (i)

    The existence results of problem (1.1) when f is indefinite and satisfies the superquadratic linear conditions;

  2. (ii)

    The phenomenon of concentration of nontrivial solutions.

In order to give positive answers to the above problems, we shall assume that the potential function \(V_{\lambda }(x)=\lambda V^{+}(x)-V^{-}(x)\), where \(V^{\pm }=\max \{\pm V,0\}\) satisfies the following conditions, which is quite different from the above cited papers.

\((V1)\) :

\(V_{\lambda }(x)\in C(\mathbb{R}^{N},\mathbb{R})\) and \(V_{\lambda }(x)\) is bounded from below;

\((V2)\) :

There exists \(b>0\) such that \(\{x\in \mathbb{R}^{N}\mid V^{+}(x)< b \}\) is nonempty and has finite measure;

\((V3)\) :

\(\Omega =\operatorname{int}\{x\in \mathbb{R}^{N}\mid V^{+}(x)=0\}\) is nonempty and has smooth boundary with \(\overline{\Omega }=\{x\in \mathbb{R}^{N}\mid V^{+}(x)=0\}\);

\((V4)\) :

There exists a constant \(\mu_{0}>1\) such that

$$ \mu_{1}(\lambda ):=\inf_{u\in H^{2}(\mathbb{R}^{N})\setminus \{0\}}\frac{ \int_{\mathbb{R}^{N}}[ \vert \Delta u \vert ^{2}+ \vert \nabla u \vert ^{2}+\lambda V^{+}(x)u ^{2}]\,dx}{\int_{\mathbb{R}^{N}}V^{-}(x)u^{2}\,dx}\geq \mu_{0}\quad \text{for all }\lambda >0. $$

Here, we point out that conditions \((V1)\)\((V3)\), which imply that \(\lambda V^{+}(x)\) represents a potential well whose depth is controlled by λ, were firstly introduced by Bartsch and Wang [3], in which the authors studied the nonlinear Schrödinger equations. For \(\lambda >0\) large, one expects to find solutions which localize near its bottom Ω. Since the work [3], there have been many papers dealing with problems with potential well in different equations, see e.g. [1, 2, 9, 10, 31]. However, to the best of our knowledge, there seems to be no result on this case of problem (1.1) with sign-changing potential. This is the reason why we explore the phenomenon of concentration of solution in this paper as well.

Remark 1.1

Inspired by [20], we impose condition \((V4)\) in this paper. Obviously, there are cases when condition \((V4)\) is easily verifiable. For example, if one takes a function \(V^{-}\in L^{\frac{2^{\ast }}{2^{\ast }-2}}(\mathbb{R}^{N})\) with \(\Vert V^{-} \Vert _{L^{\frac{2^{\ast }}{2^{\ast }-2}}}<\bar{\bar{S}}^{2}\), then a direct calculation from \((V1)\)\((V3)\), the fact \(H^{2}( \mathbb{R}^{N})\hookrightarrow H^{1}(\mathbb{R}^{N})\), and the Hölder and Sobolev inequalities show that

$$\begin{aligned} \frac{\int_{\mathbb{R}^{N}}[ \vert \Delta u \vert ^{2}+ \vert \nabla u \vert ^{2}+\lambda V ^{+}u^{2}]\,dx}{\int_{\mathbb{R}^{N}}V^{-}u^{2}\,dx} \geq & \frac{ \int_{\mathbb{R}^{N}}[ \vert \Delta u \vert ^{2}+ \vert \nabla u \vert ^{2}+\lambda V^{+}u ^{2}]\,dx}{ \Vert V^{-} \Vert _{L^{\frac{2^{\ast }}{2^{\ast }-2}}} ( \int_{ \mathbb{R}^{N}}u^{2^{\ast }}\,dx ) ^{\frac{2}{2^{\ast }}}} \\ \geq & \frac{\int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx}{ \Vert V^{-} \Vert _{L^{\frac{2^{ \ast }}{2^{\ast }-2}}} ( \int_{\mathbb{R}^{N}}u^{2^{\ast }}\,dx ) ^{\frac{2}{2^{\ast }}}} \\ \geq & \frac{\bar{\bar{S}}^{2}}{ \Vert V^{-} \Vert _{L^{\frac{2^{\ast }}{2^{ \ast }-2}}}}\quad \text{for all }\lambda \geq 0, \end{aligned}$$

which implies that

$$ \mu_{1}(\lambda )\geq \frac{\bar{\bar{S}}^{2}}{ \Vert V^{-} \Vert _{L^{\frac{2^{ \ast }}{2^{\ast }-2}}}}>1\quad \text{for all }\lambda \geq 0, $$

where \(2^{\ast }=\frac{2N}{N-2}\) for \(N\geq 3\), \(2^{\ast }=+\infty \) for \(N=1,2\) and \(\bar{\bar{S}}\) denotes the best Sobolev constant for the embedding of \(D^{1,2}(\mathbb{R}^{N})\) in \(L^{2^{\ast }}(\mathbb{R} ^{N})\).

Before stating our main results, we also need to make some assumptions for the nonlinearity f and its primitive \(F(x,u)=\int_{0}^{u}f(x,s)\,ds\).

\((F1)\) :

\(f\in C(\mathbb{R}^{N}\times \mathbb{R},\mathbb{R})\), and there exist \(c_{1}>0\), \(p\in (2,2_{\ast })\) such that

$$ \bigl\vert f(x,u) \bigr\vert \leq c_{1}\bigl(1+ \vert u \vert ^{p-1}\bigr),\quad \text{for all }(x,u)\in \bigl(\mathbb{R} ^{N}\times \mathbb{R}\bigr), $$

here and hereafter \(2_{\ast }=\frac{2N}{N-4}\) for \(N\geq 5\), \(2_{\ast }=+\infty \) for \(N<5\);

\((F2)\) :

\(\lim_{ \vert u \vert \rightarrow 0}\frac{f(x,u)}{u}=0\) uniformly for \(x\in \mathbb{R}^{N}\);

\((F3)\) :

\(\lim_{ \vert u \vert \rightarrow \infty }\frac{F(x,u)}{ \vert u \vert ^{q}}=+\infty \) uniformly for \(q\in (2,p)\) and \(x\in \mathbb{R}^{N}\);

\((F4)\) :

There exist \(\tau >2\) and \(C_{2}>0\) such that

$$ \widetilde{F}(x,u):=\frac{1}{\tau }f(x,u)-F(x,u)\rightarrow +\infty , \quad \text{as } \vert u \vert \rightarrow +\infty \text{ uniformly in } x\in \mathbb{R}^{N}, $$

and

$$ \widetilde{F}(x,u_{n})\geq -C_{2} \vert u_{n} \vert ^{2},\quad \forall x\in \mathbb{R}^{N}. $$
\((F5)\) :

\(f(x,-u)=-f(x,u)\) for all \((x,u)\in (\mathbb{R}^{N}\times \mathbb{R})\).

Remark 1.2

There are functions \(f(x,u)\) satisfying conditions \((F1)\)\((F5)\) in this paper. For example, let

$$\begin{aligned} f(x,u)= \textstyle\begin{cases} g(x)2\ln 2 \vert u \vert ^{p-2}u+\frac{u}{2} & \text{for } \vert u \vert >1, \\ g(x)2u\ln (1+ \vert u \vert )+\frac{ \vert u \vert u}{1+ \vert u \vert } & \text{for } \vert u \vert \leq 1, \end{cases}\displaystyle \end{aligned}$$

where \(2< p<2_{\ast }\) and \(g(x)\in C(\mathbb{R}^{N}\times \mathbb{R})\) is a bounded function with \(\inf_{x\in \mathbb{R}^{N}}g(x)>0\). Then

$$\begin{aligned} F(x,u)= \textstyle\begin{cases} g(x)\frac{2\ln 2}{p} \vert u \vert ^{p}+\frac{u^{2}}{4}-\frac{1}{4}+\frac{p-2}{p} \ln 2 & \text{for } \vert u \vert >1, \\ g(x)u^{2}\ln (1+ \vert u \vert ) & \text{for } \vert u \vert \leq 1. \end{cases}\displaystyle \end{aligned}$$

Hence, it is easy to check that conditions \((F1)\)\((F5)\) are satisfied.

Now, we give the main results as follows.

Theorem 1.1

Assume that \((V1)\)\((V4)\) and \((F1)\)\((F4)\) are satisfied. There exists a constant \(\Lambda >0\) such that problem (1.1) possesses a nontrivial solution for \(\lambda >\Lambda \).

Theorem 1.2

Assume that \((V1)\)\((V4)\) and \((F1)\)\((F5)\) are satisfied. There exists a constant \(\Lambda >0\) such that, for \(\lambda >\Lambda \), problem (1.1) possesses infinitely many solutions \(\{u_{k}\}\) satisfying

$$ \frac{1}{2} \int_{\mathbb{R}^{N}}\bigl[ \vert \Delta u \vert ^{2}+ \vert \nabla u \vert ^{2}+V_{ \lambda }(x)u^{2}\bigr] \,dx - \int_{\mathbb{R}^{N}} F(x,u)\,dx\rightarrow + \infty \quad \textit{as } k \rightarrow +\infty . $$

For \(\lambda \rightarrow \infty \), we have further information on the solution \(u_{\lambda }\) which is obtained in Theorem 1.1.

Theorem 1.3

Let \(u_{\lambda }\) be the solution obtained by Theorem 1.1. Then \(u_{\lambda }\rightarrow u_{0}\) in \(H^{2}(\mathbb{R}^{N})\) as \(\lambda \rightarrow \infty \), where \(u_{0}\in H^{2}_{0}(\Omega )\) is the nontrivial solution of

$$\begin{aligned} \textstyle\begin{cases} \Delta^{2} u-\Delta u- V^{-}(x)u=f(x,u), & x\in \Omega , \\ u=0, & x\in \partial \Omega . \end{cases}\displaystyle \end{aligned}$$
(1.2)

It is worth emphasizing that under conditions \((V_{1}')\) and \((V_{2}')\), motivated by Lemma 3.4 in [33], we can prove that the working space \(E\hookrightarrow L^{s}(\mathbb{R}^{N})\) is compact for any \(s\in [2,2_{\ast })\), where \(E:= \{ u\in H^{2}( \mathbb{R}^{N}):\int_{\mathbb{R}^{N}} V^{+}(x)u^{2}\,dx<\infty \} \). Hence, the corresponding results in the present paper have been obtained by using variational techniques in a standard way [23, 25]. However, under conditions \((V1)\) and \((V2)\), the embedding lacks the compactness. This leads to a difficulty in using variational methods to get solutions of problem (1.1) since some techniques in compact cases do not work. To overcome this obstacle we have to search for other methods. Motivated by Brezis–Lieb lemma [5], we prove that the functional \(I_{\lambda }\) and its derivative \(I_{\lambda }'\) possess BL-splitting property (see Lemma 3.2). This important proposition paves the way for us to verify the boundedness of a Cerami sequence. Also, the term \(\int_{\mathbb{R}^{N}}V^{-}(x)u^{2}\,dx\) is an issue for employing the variational methods. To get over this difficulty, some new inequalities are established. In addition, we consider the problem with more general potential, which includes the positive case in the aforementioned references. Moreover, from conditions \((F1)\)\((F4)\), one can see that the nonlinearity \(f(x,u)\) and its primitive \(F(x,u)\) may change signs. That are the reasons why we say the sign-changing potential and indefinite nonlinearity on the title. Therefore, the corresponding results in the related papers are extended.

The remainder of this paper is organized as follows: In Sect. 2, some preliminaries and variational setting are presented; in Sect. 3, some important lemmas are given while the proofs of the main results are presented in Sect. 4.

Preliminaries and variational setting

In the present paper, we use the following notations:

  • For any \(\rho >0\) and for any \(z\in \mathbb{R}^{N}\), \(B_{\rho }(z)\) denotes the ball of radius ρ centered at z.

  • C and \(C_{i}\) denote various positive constants, which may vary from line to line.

  • → denotes the strong convergence and denotes the weak convergence.

  • \(o(1)\) denotes any quantity which tends to zero when \(n\rightarrow \infty \).

  • If we take a subsequence of a sequence \(\{u_{n}\}\), we shall denote it again by \(\{u_{n}\}\).

  • \(L^{q}(\mathbb{R}^{N})\) denotes the weighted space of measurable functions \(u:\mathbb{R}^{N}\rightarrow \mathbb{R}\) satisfying

    $$ \Vert u \Vert _{q}= \biggl( \int_{\mathbb{R}^{N}} \vert u \vert ^{q}\,dx \biggr) ^{\frac{1}{q}}< \infty . $$
  • \(H^{2}(\mathbb{R}^{N}):=W^{2,2}(\mathbb{R}^{N})\) denotes the space with the inner product and norm

    $$ \langle u,v\rangle_{H^{2}}= \int_{\mathbb{R}^{N}}[\Delta u\Delta v+ \nabla u\nabla v+ uv]\,dx,\quad \quad \Vert u \Vert _{H^{2}}=\langle u,u\rangle_{H^{2}} ^{\frac{1}{2}}. $$

Followed by [31], set

$$ E:= \biggl\{ u\in H^{2}\bigl(\mathbb{R}^{N}\bigr): \int_{\mathbb{R}^{N}} V^{+}(x)u ^{2}\,dx< \infty \biggr\} $$

be equipped with the inner product and norm

$$ \langle u,v\rangle_{E}= \int_{\mathbb{R}^{N}}\bigl[\Delta u\Delta v+\nabla u\nabla v+ V^{+}(x)uv\bigr]\,dx,\quad\quad \Vert u \Vert _{E}= \langle u,u\rangle_{E}^{ \frac{1}{2}}. $$

Under conditions \((V1)\) and \((V2)\), Lemma 2.1 in [24] shows that \(E\hookrightarrow H^{2}(\mathbb{R}^{N})\) is continuous, i.e., there exists a positive constant \(C_{E}\) such that

$$ \Vert u \Vert _{H^{2}}\leq C_{E} \Vert u \Vert _{E}. $$
(2.1)

For \(\lambda >0\), we also need the following inner product and norm:

$$ \langle u,v\rangle_{\lambda }= \int_{\mathbb{R}^{N}}\bigl[\Delta u\Delta v+ \nabla u\nabla v+ \lambda V^{+}(x)uv\bigr]\,dx,\quad\quad \Vert u \Vert _{\lambda }= \langle u,u\rangle_{\lambda }^{\frac{1}{2}}. $$

Obviously, \(\Vert u \Vert _{\lambda }\geq \Vert u \Vert _{E}\) for all \(\lambda \geq 1\). Furthermore, it follows from condition \((V4)\) that

$$ \int_{\mathbb{R}^{N}}\bigl[ \vert \Delta u \vert ^{2}+ \vert \nabla u \vert ^{2}+V_{\lambda }(x)u ^{2}\bigr] \,dx\geq \frac{\mu_{0}-1}{\mu_{0}} \Vert u \Vert _{\lambda }^{2}\quad \text{for all }\lambda \geq 0. $$
(2.2)

Set \(E_{\lambda }=(E, \Vert u \Vert _{\lambda })\). For any \(r\in [2,2_{\ast }]\) and \(\lambda \geq 1\), applying (2.1), \((V1)\), \((V2)\), the Hölder and Sobolev inequalities yields that

$$\begin{aligned} \int_{\mathbb{R}^{N}} \vert u \vert ^{r} \,dx \leq & \biggl( \int_{\{V^{+}(x)\geq b\}} \vert u \vert ^{2} \,dx+ \int_{\{V^{+}(x)< b \}} \vert u \vert ^{2} \,dx \biggr) ^{\frac{2_{\ast }-r}{2_{\ast }-2}} \biggl( \int_{ \mathbb{R}^{N}} \vert u \vert ^{2_{\ast }}\,dx \biggr) ^{\frac{r-2}{2_{\ast }-2}} \\ \leq & \biggl[ \frac{1}{\lambda b} \int_{\{V^{+}(x)\geq b\}}\lambda V ^{+}(x) \vert u \vert ^{2} \,dx+ \bigl\vert \bigl\{ V^{+}< b\bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{\ast }}} \biggl( \int _{\mathbb{R}^{N}} \vert u \vert ^{2_{\ast }} \,dx \biggr) ^{\frac{2}{2_{\ast }}} \biggr] ^{\frac{2_{\ast }-r}{2_{\ast }-2}} \\ & {} \cdot \biggl[ \bar{S}^{-2_{\ast }} \biggl( \int_{\mathbb{R}^{N}}\bigl[ \vert \triangle u \vert ^{2}+ \vert \nabla u \vert ^{2}+u^{2}\bigr] \,dx \biggr) ^{\frac{2_{\ast }}{2}} \biggr] ^{\frac{r-2}{2_{\ast }-2}} \\ \leq & \biggl( \frac{1}{\lambda b} \Vert u \Vert _{\lambda }^{2} + \bigl\vert \bigl\{ V^{+}(x)< b \bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{\ast }}} \bar{S}^{-2}C_{E}^{2} \Vert u \Vert _{\lambda }^{2} \biggr) ^{\frac{2_{\ast }-r}{2_{\ast }-2}} \bigl( \bar{S}^{-2_{ \ast }}C_{E}^{2_{\ast }} \Vert u \Vert _{\lambda }^{2_{\ast }} \bigr) ^{\frac{r-2}{2_{ \ast }-2}} \\ =& \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{ \ast }}}\bar{S}^{-2}C_{E}^{2} \biggr) ^{\frac{2_{\ast }-r}{2_{\ast }-2}} \bigl( \bar{S}^{-2_{\ast }}C_{E}^{2_{\ast }} \bigr) ^{\frac{r-2}{2_{ \ast }-2}} \Vert u \Vert _{\lambda }^{r}, \end{aligned}$$
(2.3)

which means that \(E_{\lambda }\hookrightarrow L^{r}(\mathbb{R}^{N})\) is continuous for \(r\in [2,2_{\ast }]\), where is the best Sobolev constant for the imbedding of \(H^{2}(\mathbb{R}^{N})\hookrightarrow L ^{2_{\ast }}(\mathbb{R}^{N})\).

Define a functional \(I_{\lambda }\) on \(E_{\lambda }\) by

$$\begin{aligned} I_{\lambda }(u) =&\frac{1}{2} \int_{\mathbb{R}^{N}}\bigl[ \vert \Delta u \vert ^{2}+ \vert \nabla u \vert ^{2}+V_{\lambda }(x)u^{2}\bigr] \,dx- \int_{\mathbb{R}^{N}} F(x,u)\,dx \\ =&\frac{1}{2} \Vert u \Vert _{\lambda }^{2}- \frac{1}{2} \int_{\mathbb{R}^{N}}V ^{-}(x)u^{2}\,dx- \int_{\mathbb{R}^{N}} F(x,u)\,dx,\quad \forall u\in E _{\lambda }. \end{aligned}$$
(2.4)

Followed by [24], the functional \(I_{\lambda }\) is of class \(C^{1}(E_{\lambda },\mathbb{R})\) and

$$ \bigl\langle I'_{\lambda }(u),\varphi \bigr\rangle = \int_{\mathbb{R}^{N}}\bigl[ \triangle u\triangle \varphi +\nabla u\nabla \varphi + V_{\lambda }(x)u \varphi \bigr] \,dx - \int_{\mathbb{R}^{N}}f(x,u)\varphi \,dx,\quad \forall u, \varphi \in E_{\lambda }. $$
(2.5)

Hence, if \(u\in E_{\lambda }\) is a critical point of \(I_{\lambda }\), then u is a solution of problem (1.1).

We shall end this section by giving the following definition and propositions which are applied to prove the main results.

Definition 2.1

Let X be a real Banach space and \(I\in C^{1}(X,\mathbb{R})\). For some \(c\in \mathbb{R}\), we say I satisfies the \((C)_{c}\) condition if any sequence \(\{u_{n}\}\subset X\) such that \(I(u_{n})\rightarrow c\) and \(\Vert I'(u_{n}) \Vert (1+ \Vert u_{n} \Vert ) \rightarrow 0\) as \(n\rightarrow \infty \) has a convergent subsequence.

Proposition 2.1

(Mountain Pass Theorem [18])

Let X be a real Banach space, \(I\in C^{1}(X, \mathbb{R})\) satisfies the \((C)_{c}\) condition for any \(c>0\), \(I(0)=0\) and

  1. (i)

    there exist \(\rho , \alpha >0\) such that \(I| _{\partial B_{ \rho }}\geq \alpha \);

  2. (ii)

    there exists \(e\in E\setminus B_{\rho }\) such that \(I(e)\leq 0\).

Then I has a critical value \(c\geq \alpha \).

Proposition 2.2

(Symmetric Mountain Pass Theorem [18])

Let X be an infinite dimensional Banach space, and let \(I\in C^{1}(X,\mathbb{R})\) be even, satisfy \((C)_{c}\) condition and \(I(0)=0\). If \(X=V\oplus W\), where V is finite dimensional, and I satisfies

\((A_{1})\) :

there are constants \(\rho , \alpha >0\) such that \(I| _{\partial B_{\rho }\cap W}\geq \alpha \), and

\((A_{2})\) :

for each finite dimensional subspace \(\widetilde{X}\subset X\), there is \(R=R(\widetilde{X})\) such that \(I\leq 0\) on \(\widetilde{X}\setminus B_{R}(\widetilde{X})\).

Then I possesses an unbounded sequence of critical values.

Some lemmas

To verify the main results, we need the following lemmas first.

Lemma 3.1

Assume that \((V1)\)\((V4)\) and \((F1)\)\((F3)\) hold. Let \(e\in E_{\lambda }\) with \(e\neq 0\). Then

  1. (i)

    there exist \(\rho , \alpha >0\) such that \(I_{\lambda }| _{\partial B_{\rho }}\geq \alpha \);

  2. (ii)

    \(I_{\lambda }(te)\rightarrow -\infty \) as \(t\rightarrow \infty \).

Proof

From \((F1)\) and \((F2)\), for any \(\varepsilon >0\), there exists \(C(\varepsilon )>0\) such that

$$ \bigl\vert f(x,u) \bigr\vert \leq \varepsilon \vert u \vert +C(\varepsilon ) \vert u \vert ^{p-1}, $$
(3.1)

then

$$ \bigl\vert F(x,u) \bigr\vert \leq \varepsilon \vert u \vert ^{2}+C(\varepsilon ) \vert u \vert ^{p}. $$
(3.2)

It deduces from (2.2), (2.3), (2.4), and (3.2) that

$$\begin{aligned} I_{\lambda }(u) =&\frac{1}{2} \int_{\mathbb{R}^{N}}\bigl[ \vert \Delta u \vert ^{2}+ \vert \nabla u \vert ^{2}+V_{\lambda }(x)u^{2}\bigr] \,dx- \int_{\mathbb{R}^{N}} F(x,u)\,dx \\ \geq &\frac{\mu_{0}-1}{2\mu_{0}} \Vert u \Vert _{\lambda }^{2} - \int_{\mathbb{R}^{N}} F(x,u)\,dx \\ \geq &\frac{\mu_{0}-1}{2\mu_{0}} \Vert u \Vert _{\lambda }^{2} - \int_{\mathbb{R}^{N}} \bigl[\varepsilon \vert u \vert ^{2}+C( \varepsilon ) \vert u \vert ^{p}\bigr]\,dx \\ \geq & \biggl[ \frac{\mu_{0}-1}{2\mu_{0}}-\varepsilon \biggl( \frac{1}{ \lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{\ast }}} \bar{S} ^{-2}C_{E}^{2} \biggr) \biggr] \Vert u \Vert _{\lambda }^{2}-CC(\varepsilon ) \Vert u \Vert _{\lambda }^{p}. \end{aligned}$$
(3.3)

Therefore, the conclusion (i) follows from taking \(0<\varepsilon <\frac{ \mu_{0}-1}{2\mu_{0}} ( \frac{1}{\lambda b} + \vert \{V^{+}(x)< b\} \vert ^{\frac{2_{ \ast }-2}{2_{\ast }}}\bar{S}^{-2} C_{E}^{2} ) ^{-1}\) and choosing \(\Vert u \Vert _{\lambda }=\rho \) sufficiently small since \(p>2\).

Next, we shall show the conclusion (ii). From \((F3)\), for any \(M>0\), there exists \(\delta =\delta (M)>0\) such that

$$ F(x,u)\geq M \vert u \vert ^{q}\quad \text{for all }x\in \mathbb{R}^{N} \text{and } \vert u \vert >\delta . $$
(3.4)

By \((F1)\) and \((F2)\), there exists \(M_{1}=M_{1}(M)>0\) such that

$$ \frac{ \vert f(x,u)u \vert }{ \vert u \vert ^{2}}\leq M_{1}\quad \text{for all } x\in \mathbb{R}^{N}\text{and } 0< \vert u \vert \leq \delta , $$

which combining with the mean value theorem gives that

$$ \bigl\vert F(x,u) \bigr\vert \leq \frac{M_{1}}{2} \vert u \vert ^{2}\quad \text{for all } x\in \mathbb{R}^{N} \text{and } 0< \vert u \vert \leq \delta . $$
(3.5)

Denote \(\bar{M}=M \vert \delta \vert ^{q-2}+\frac{M_{1}}{2}\). Then (3.4) and (3.5) imply that

$$ F(x,u)\geq M \vert u \vert ^{q}-\bar{M} \vert u \vert ^{2}\quad \text{for all } (x,u)\in \bigl( \mathbb{R}^{N} \times \mathbb{R}\bigr). $$
(3.6)

Therefore, for any given \(e\in E_{\lambda }\), it follows from (2.4) and (3.6) that

$$\begin{aligned} I_{\lambda }(te) =&\frac{t^{2}}{2} \int_{\mathbb{R}^{N}}\bigl[ \vert \Delta e \vert ^{2}+ \vert \nabla e \vert ^{2}+V_{\lambda }(x)e^{2}\bigr] \,dx - \int_{\mathbb{R}^{N}} F(x,te)\,dx \\ \leq &\frac{t^{2}}{2} \Vert e \Vert _{\lambda }^{2} - \int_{\mathbb{R}^{N}}\bigl[t ^{q}M \vert e \vert ^{q}-t^{2}\bar{M} \vert e \vert ^{2}\bigr]\,dx \\ \leq &\frac{t^{2}}{2} \Vert e \Vert _{\lambda }^{2} -t^{q}M \Vert e \Vert _{q}^{q}+t^{2} \bar{M} \Vert e \Vert _{2}^{2} \\ \rightarrow & -\infty ,\quad \text{as }t\rightarrow \infty , \end{aligned}$$

which means that the conclusion (ii) holds. This completes the proof. □

Lemma 3.2

Suppose that \((V1)\)\((V4)\), \((F1)\) and \((F2)\) are satisfied. Moreover, if \(u_{n}\rightharpoonup u\) in \(E_{\lambda }\), then passing to a subsequence, the following conclusions

$$ I_{\lambda }(u_{n}-u)=I_{\lambda }(u_{n})-I_{\lambda }(u)+o(1) $$
(3.7)

and

$$ I_{\lambda }'(u_{n}-u)=I_{\lambda }'(u_{n})-I_{\lambda }'(u)+o(1) $$
(3.8)

as \(n\rightarrow \infty \) are satisfied.

Proof

It follows from the assumption \(u_{n}\rightharpoonup u\) in \(E_{\lambda }\) that \(\langle u_{n},u\rangle_{\lambda }\rightarrow \langle u,u\rangle_{\lambda }\) as \(n\rightarrow \infty \), which yields that

$$\begin{aligned} \Vert u_{n} \Vert _{\lambda }^{2} =&\langle u_{n},u_{n}\rangle_{\lambda } \\ =& \langle u_{n}-u,u_{n}-u\rangle_{\lambda }+\langle u,u_{n} \rangle_{\lambda }+\langle u_{n}-u,u \rangle_{\lambda } \\ =& \Vert u_{n}-u \Vert _{\lambda }^{2}+ \Vert u \Vert _{\lambda }^{2}+o(1). \end{aligned}$$

For all \(\varphi \in E_{\lambda }\), it is clear that

$$ \langle u_{n},\varphi \rangle_{\lambda }=\langle u_{n}-u,\varphi \rangle_{\lambda }+\langle u,\varphi \rangle_{\lambda }. $$

Note that conditions \((V1)\) and \((V2)\) imply that \(V^{-}(x)\geq 0\) for all \(x\in \mathbb{R}^{N}\) and \(V^{-}(x)\in L^{\infty }(\mathbb{R}^{N})\). Moreover, from condition \((V2)\) it follows that \(\{V^{+}(x)=0\}\) has finite measure, which implies that \(\{V^{-}(x)>0\}\) has finite measure. Hence, applying the facts \(u_{n}\rightharpoonup u\) in \(E_{\lambda }\) and \(u_{n}\rightarrow u\) in \(L_{\mathrm {loc}}^{2}(\mathbb{R}^{N})\) gives that

$$\begin{aligned} \biggl\vert \int_{\mathbb{R}^{N}}V^{-}(x) (u_{n}-u)^{2} \,dx \biggr\vert =& \biggl\vert \int _{\operatorname {supp}V^{-}}V^{-}(x) (u_{n}-u)^{2} \,dx \biggr\vert \\ \leq & \bigl\Vert V^{-} \bigr\Vert _{\infty } \int_{\operatorname {supp}V^{-}}(u_{n}-u)^{2}\,dx\rightarrow 0 \end{aligned}$$
(3.9)

and

$$\begin{aligned} \biggl\vert \int_{\mathbb{R}^{N}}V^{-}(x) \bigl(u_{n}^{2}-u^{2} \bigr)\,dx \biggr\vert =& \biggl\vert \int _{\operatorname {supp}V^{-}}V^{-}(x) (u_{n}-u) (u_{n}+u)\,dx \biggr\vert \\ \leq & \bigl\Vert V^{-} \bigr\Vert _{\infty } \int_{\operatorname {supp}V^{-}} \vert u_{n}-u \vert \vert u_{n}+u \vert \,dx \\ \leq & \bigl\Vert V^{-} \bigr\Vert _{\infty } \biggl( \int_{\operatorname {supp}V^{-}} \vert u_{n}-u \vert ^{2}\,dx \biggr) ^{\frac{1}{2}} \biggl( \int_{\operatorname {supp}V^{-}} \vert u_{n}+u \vert ^{2}\,dx \biggr) ^{ \frac{1}{2}} \\ \rightarrow & 0. \end{aligned}$$
(3.10)

An easy calculation from (3.9) and (3.10) shows that

$$ \int_{\mathbb{R}^{N}}V^{-}(x) (u_{n}-u)^{2} \,dx = \int_{\mathbb{R}^{N}}V ^{-}(x)u_{n}^{2}\,dx- \int_{\mathbb{R}^{N}}V^{-}(x)u^{2}\,dx+o(1). $$

Similarly, for any \(\varphi \in E_{\lambda }\), one can also obtain that

$$ \int_{\mathbb{R}^{N}}V^{-}(x) (u_{n}-u)\varphi \,dx = \int_{\mathbb{R} ^{N}}V^{-}(x)u_{n}\varphi \,dx- \int_{\mathbb{R}^{N}}V^{-}(x)u\varphi \,dx+o(1). $$

Therefore, to prove (3.7) and (3.8), it suffices to check that

$$ \int_{\mathbb{R}^{N}}\bigl[F(x,u_{n})-F(x,u_{n}-u)-F(x,u) \bigr]\,dx=o(1) $$
(3.11)

and

$$ \sup_{\varphi \in E_{\lambda }, \Vert \varphi \Vert _{\lambda }=1} \int_{\mathbb{R}^{N}}\bigl[f(x,u_{n})-f(x,u_{n}-u)-f(x,u) \bigr]\varphi \,dx=o(1). $$
(3.12)

Here, we only show (3.11) since the verification of (3.12) is similar. Inspired by [5, 24], let \(w_{n}:=u_{n}-u\). Then \(w_{n}\rightharpoonup 0\) in \(E_{\lambda }\) and \(w_{n}(x)\rightarrow 0\) a.e. \(x\in \mathbb{R}^{N}\). It follows from (3.1) that

$$\begin{aligned} \bigl\vert F(x,w_{n}+u)-F(x,w_{n}) \bigr\vert \leq & \int_{0}^{1} \bigl\vert f(x,w_{n}+su)u \bigr\vert \,ds \\ \leq & \int_{0}^{1}\bigl(\varepsilon \vert w_{n}+su \vert \vert u \vert +C(\varepsilon ) \vert w_{n}+su \vert ^{p-1} \vert u \vert \bigr)\,ds \\ \leq & C \bigl( \varepsilon \vert w_{n} \vert \vert u \vert +\varepsilon u^{2}+C(\varepsilon ) \vert w_{n} \vert ^{p-1} \vert u \vert +C(\varepsilon ) \vert u \vert ^{p} \bigr) . \end{aligned}$$

Then Young’s inequality implies that

$$ \bigl\vert F(x,w_{n}+u)-F(x,w_{n}) \bigr\vert \leq C \bigl( \varepsilon \vert w_{n} \vert ^{2}+\varepsilon \vert u \vert ^{2}+C(\varepsilon ) \vert w_{n} \vert ^{p}+C(\varepsilon ) \vert u \vert ^{p} \bigr) , $$

which combining with (3.2) yields that

$$ \bigl\vert F(x,w_{n}+u)-F(x,w_{n})-F(x,u) \bigr\vert \leq C \bigl( \varepsilon \vert w_{n} \vert ^{2}+ \varepsilon \vert u \vert ^{2}+C(\varepsilon ) \vert w_{n} \vert ^{p}+C(\varepsilon ) \vert u \vert ^{p} \bigr) , \quad n\in \mathbb{N}. $$

Let

$$ H_{n}(x):=\max \bigl\{ \bigl\vert F(x,w_{n}+u)-F(x,w_{n})-F(x,u) \bigr\vert - C\varepsilon \bigl( \vert w _{n} \vert ^{2}+ \vert w_{n} \vert ^{p}\bigr),0\bigr\} . $$

Then

$$ 0\leq H_{n}(x)\leq C\varepsilon \bigl( \vert w_{n} \vert ^{2}+ \vert u \vert ^{p}\bigr)\in L^{1} \bigl( \mathbb{R}^{N}\bigr). $$

Thus, the Lebesgue dominated convergence theorem implies that

$$ \int_{\mathbb{R}^{N}} H_{n}(x)\,dx\rightarrow 0\quad \text{as }n \rightarrow \infty . $$
(3.13)

Furthermore, by the definition of \(H_{n}(x)\), we have

$$ \bigl\vert F(x,w_{n}+u)-F(x,w_{n})-F(x,u) \bigr\vert \leq C\varepsilon \bigl( \vert w_{n} \vert ^{2}+ \vert w_{n} \vert ^{p}\bigr)+H _{n}(x),\quad \forall n\in \mathbb{N}, $$

which together with (2.3), (3.1), and (3.13) shows that

$$\begin{aligned} \int_{\mathbb{R}^{N}} \bigl\vert F(x,w_{n}+u)-F(x,w_{n})-F(x,u) \bigr\vert \,dx \leq & C \varepsilon \bigl( \Vert w_{n} \Vert _{2}^{2}+ \Vert w_{n} \Vert _{p}^{p}\bigr)+\varepsilon \\ \leq & C\varepsilon \bigl( \Vert w_{n} \Vert _{\lambda }^{2}+ \Vert w_{n} \Vert _{\lambda } ^{p}\bigr)+\varepsilon \\ \leq & C\varepsilon \end{aligned}$$

for n sufficiently large. Hence, (3.11) holds. This completes the proof. □

Lemma 3.3

Assume that \((V1)\)\((V4)\) and \((F1)\)\((F4)\) hold. Then any \((C)_{c}\) sequence of \(I_{\lambda }\) is bounded in \(E_{\lambda }\).

Proof

Let \(\{u_{n}\}\subset E_{\lambda }\) be a \((C)_{c}\) sequence of \(I_{\lambda }\), that is,

$$ I_{\lambda }(u_{n})\rightarrow c,\quad\quad \bigl\Vert I'_{\lambda }(u_{n}) \bigr\Vert \bigl(1+ \Vert u _{n} \Vert _{\lambda }\bigr)\rightarrow 0,\quad \text{as }n\rightarrow \infty . $$
(3.14)

To prove the boundedness of \(\{u_{n}\}\) in \(E_{\lambda }\), arguing by contradiction, we suppose that \(\Vert u_{n} \Vert _{\lambda }\rightarrow \infty \) as \(n\rightarrow \infty \). Hence, for n sufficiently large, there exists a positive constant \(C_{1}\) such that

$$\begin{aligned} C_{1}+ \Vert u_{n} \Vert _{\lambda } \geq & I_{\lambda }(u_{n})-\frac{1}{\tau } \bigl\langle I'_{\lambda }(u_{n}),u_{n}\bigr\rangle \\ \geq &\frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}-\frac{1}{\tau } \biggr) \Vert u_{n} \Vert _{\lambda }^{2}+ \int_{\mathbb{R}^{N}}\widetilde{F}(x,u)\,dx, \end{aligned}$$

which implies that

$$ C_{1}\geq \frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}-\frac{1}{\tau } \biggr) \Vert u_{n} \Vert _{\lambda }^{2}- \Vert u_{n} \Vert _{\lambda }+ \int_{\mathbb{R}^{N}} \widetilde{F}(x,u)\,dx\geq \int_{\mathbb{R}^{N}}\widetilde{F}(x,u)\,dx $$
(3.15)

for n sufficiently large. Set \(w_{n}=\frac{u_{n}}{ \Vert u_{n} \Vert _{\lambda }}\). Then \(\Vert w_{n} \Vert _{\lambda }=1\). For \(\lambda \geq 1\), noting that

$$ \frac{\int_{\mathbb{R}^{N}}[ \vert \Delta u_{n} \vert ^{2}+ \vert \nabla u_{n} \vert ^{2}+V _{\lambda }(x)u_{n}^{2}]\,dx}{ \Vert u_{n} \Vert _{\lambda }^{q}}\leq \frac{ \int_{\mathbb{R}^{N}}[ \vert \Delta u_{n} \vert ^{2}+ \vert \nabla u_{n} \vert ^{2}+V^{+}(x)u _{n}^{2}]\,dx}{ \Vert u_{n} \Vert _{\lambda }^{q}} \leq \frac{1}{ \Vert u_{n} \Vert _{ \lambda }^{q-2}} $$

and

$$ \frac{\langle I'_{\lambda }(u_{n}),u_{n}\rangle }{ \Vert u_{n} \Vert _{\lambda }^{q}}=\frac{\int_{\mathbb{R}^{N}}[ \vert \Delta u_{n} \vert ^{2}+ \vert \nabla u_{n} \vert ^{2}+V _{\lambda }(x)u_{n}^{2}]\,dx}{ \Vert u_{n} \Vert _{\lambda }^{q}} - \int_{\mathbb{R}^{N}}\frac{f(x,u_{n})u_{n}}{ \Vert u_{n} \Vert _{\lambda }^{q}}\,dx, $$

one has

$$ \lim_{n\rightarrow +\infty } \int_{\mathbb{R}^{N}}\frac{f(x,u_{n})u _{n}}{ \Vert u_{n} \Vert _{\lambda }^{q}}\,dx= 0. $$

Note that \(\Vert w_{n} \Vert _{\lambda }=1\). Then up to a subsequence, we may assume \(w_{n}\rightharpoonup w\) in \(E_{\lambda }\) and \(w_{n}\rightarrow w\) a.e. \(\mathbb{R}^{N}\). Set

$$ \mathcal{B}=\bigl\{ x\in \mathbb{R}^{N}\mid w(x)\neq 0\bigr\} . $$

Now, we show that \(\operatorname{meas}(\mathcal{B})=0\). Otherwise, \(\vert u_{n}(x) \vert \rightarrow +\infty \) for a.e. \(x\in \mathcal{B}\). For any constant \(M_{1}>0\), \((F1)\), \((F2)\), and \((F3)\) imply that

$$ f(x,u_{n})u_{n}\geq M_{1} \vert u_{n} \vert ^{q}-C(M_{1}) \vert u_{n} \vert ^{2}\quad \text{for all }x\in \mathbb{R}^{N}. $$

Then

$$ \int_{\mathbb{R}^{N}}\frac{f(x,u_{n})u_{n}}{ \Vert u_{n} \Vert _{\lambda }^{q}}\,dx \geq M_{1} \Vert w_{n} \Vert _{q}^{q}-C(M_{1}) \frac{ \Vert w_{n} \Vert _{2}^{2}}{ \Vert u_{n} \Vert _{\lambda }^{q-2}}. $$

Consequently,

$$ 0=\lim_{n\rightarrow +\infty } \int_{\mathbb{R}^{N}}\frac{f(x,u_{n})u _{n}}{ \Vert u_{n} \Vert _{\lambda }^{q}}\,dx\geq M_{1} \Vert w \Vert _{q}^{q}=M_{1} \int_{\mathcal{B}} \vert w \vert ^{q}\,dx>0, $$

which is absurd. Hence, \(\operatorname{meas}(\mathcal{B})=0\). Therefore, \(w(x)=0\) for a.e. \(x\in \mathbb{R}^{N}\). Then it follows from (2.2), (2.4), (3.14), and \((F4)\) that

$$\begin{aligned} \frac{c+o(1)}{ \Vert u_{n} \Vert _{\lambda }^{2}} =&\frac{1}{ \Vert u_{n} \Vert _{\lambda }^{2}} \biggl[ I_{\lambda }(u_{n})- \frac{1}{\tau }\bigl\langle I'_{\lambda }(u _{n}),u_{n}\bigr\rangle \biggr] \\ \geq &\frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}-\frac{1}{\tau } \biggr) + \int_{\mathbb{R}^{N}}\frac{\widetilde{F}(x,u_{n})}{ \Vert u_{n} \Vert _{\lambda }^{2}}\,dx \\ \geq &\frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}-\frac{1}{\tau } \biggr) -C _{2} \Vert w_{n} \Vert _{2}^{2}, \end{aligned}$$

which implies \(0\geq \frac{\mu_{0}-1}{\mu_{0}} ( \frac{1}{2}-\frac{1}{ \tau } ) \) as \(n\rightarrow \infty \). This is a contradiction with \(\frac{\mu_{-}1}{\mu_{0}} ( \frac{1}{2}-\frac{1}{\tau } ) >0\). Therefore, \(\{u_{n}\}\) is bounded in \(E_{\lambda }\). We complete the proof. □

Lemma 3.4

Assume that \((V1)\)\((V4)\), \((F1)\), \((F2)\), and \((F4)\) are satisfied. Then there exists a constant Λ such that any \((C)_{c}\) sequence of \(I_{\lambda }\) possesses a convergent subsequence in \(E_{\lambda }\) for \(\lambda >\Lambda \).

Proof

Let \(\{u_{n}\}\) be a \((C)_{c}\) sequence. By the boundedness of \(\{u_{n}\}\), there exist a subsequence \(\{u_{n}\}\) and \(u_{0}\) such that

$$\begin{aligned}& u_{n}\rightharpoonup u_{0},\quad \text{in }E_{\lambda }; \\& u_{n}\rightarrow u_{0},\quad \text{in }L_{\mathrm {loc}}^{p} \bigl(\mathbb{R}^{N}\bigr), \text{for } 2\leq p< 2_{\ast }; \\& u_{n}\rightarrow u_{0},\quad \text{a.e. } \mathbb{R}^{N}. \end{aligned}$$

In what follows, we shall prove that \(u_{n}\rightarrow u_{0}\) in \(E_{\lambda }\). Let \(v_{n}:=u_{n}-u_{0}\). Then \(v_{n}\rightharpoonup 0\) in \(E_{\lambda }\). It deduces from \((V2)\) that

$$\begin{aligned} \int_{\mathbb{R}^{N}}v_{n}^{2}\,dx =& \int_{\{x\in \mathbb{R}^{N}:V^{(}x)+\geq b\}}v_{n}^{2}\,dx+ \int_{\{x\in \mathbb{R}^{N}:V^{+}(x)< b\}}v_{n}^{2}\,dx \\ \leq & \frac{1}{\lambda b} \int_{\mathbb{R}^{N}}\lambda V^{+}(x)v_{n} ^{2}\,dx+ \int_{\{x\in \mathbb{R}^{N}:V^{+}< b\}}v_{n}^{2}\,dx \\ \leq & \frac{1}{\lambda b} \Vert v_{n} \Vert _{\lambda }^{2}+o(1). \end{aligned}$$
(3.16)

Set

$$ \Lambda_{0}:=\max \biggl\{ \frac{C_{2}}{b} \biggl[ \biggl( \frac{1}{2}-\frac{1}{ \tau } \biggr) \frac{\mu_{0}-1}{\mu_{0}} \biggr] ^{-1},1 \biggr\} , $$

where \(C_{2}\) is defined in \((F4)\). Then, for \(\lambda >\Lambda_{0}\), a direct calculation from (2.2), (3.16), and the Hölder and Sobolev inequalities gives that

$$\begin{aligned} \int_{\mathbb{R}^{N}} \vert v_{n} \vert ^{r}\,dx \leq & \biggl( \int_{\mathbb{R}^{N}} \vert v _{n} \vert ^{2}\,dx \biggr) ^{\frac{2_{\ast }-r}{2_{\ast }-2}} \biggl( \int_{ \mathbb{R}^{N}} \vert v_{n} \vert ^{2_{\ast }}\,dx \biggr) ^{\frac{r-2}{2_{\ast }-2}} \\ \leq & \biggl( \frac{1}{\lambda b} \Vert v_{n} \Vert _{\lambda }^{2} \biggr) ^{\frac{2_{ \ast }-r}{2_{\ast }-2}} \biggl[ \bar{S}^{-2_{\ast }}C_{E}^{2} \biggl( \int_{ \mathbb{R}^{N}}\bigl[ \vert \Delta u \vert ^{2}+ \vert \nabla u \vert ^{2}+ \vert v_{n} \vert ^{2}\bigr]\,dx \biggr) ^{\frac{2_{\ast }}{2}} \biggr] ^{\frac{r-2}{2_{\ast }-2}} \\ &{}+o(1) \\ \leq & \biggl( \frac{1}{\lambda b} \biggr) ^{ \frac{2_{\ast }-r}{2_{\ast }-2}} \bigl( \bar{S}^{-2_{\ast }}C_{E}^{2_{ \ast }} \bigr) ^{\frac{r-2}{2_{\ast }-2}} \Vert v_{n} \Vert _{\lambda }^{r}+o(1). \end{aligned}$$
(3.17)

Moreover, for any given \(c>0\), let \(M_{2}:=c-I_{\lambda }(u_{0})\), then there exists \(C_{3}>0\) such that \(C_{3}>M_{2}\). So, combining \((F4)\), (3.17), and Lemma 3.2 gives that

$$\begin{aligned} C_{3} \geq & c-I_{\lambda }(u_{0})=I_{\lambda }(v_{n})+o(1)=I_{\lambda }(v_{n})- \frac{1}{\tau }\bigl\langle I_{\lambda }'(v_{n}),v_{n} \bigr\rangle +o(1) \\ \geq & \frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}-\frac{1}{\tau } \biggr) \Vert v_{n} \Vert _{\lambda }^{2}+ \int_{\mathbb{R}^{N}}\widetilde{F}(x,v_{n})\,dx+o(1) \\ \geq & \frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}-\frac{1}{\tau } \biggr) \Vert v_{n} \Vert _{\lambda }^{2}-C_{2} \int_{\mathbb{R}^{N}} \vert v_{n} \vert ^{2} \,dx+o(1) \\ \geq & \biggl[ \frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}-\frac{1}{ \tau } \biggr) -\frac{C_{2}}{\lambda b} \biggr] \Vert v_{n} \Vert _{\lambda }^{2}+o(1). \end{aligned}$$

This means that

$$ \Vert v_{n} \Vert _{\lambda }^{2}\leq \biggl[ \frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}-\frac{1}{ \tau } \biggr) - \frac{C_{2}}{\lambda b} \biggr] ^{-1}C_{3}+o(1)\quad \text{for } \lambda >\Lambda_{0}, $$

which together with (2.3) yields that

$$\begin{aligned} \int_{\mathbb{R}^{N}} \vert v_{n} \vert ^{r} \,dx \leq & \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b \bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{\ast }}}\bar{S}^{-2}C_{E}^{2} \biggr) ^{\frac{2_{\ast }-r}{2_{\ast }-2}} \bigl( \bar{S}^{-2_{\ast }}C_{E} ^{2_{\ast }} \bigr) ^{\frac{r-2}{2_{\ast }-2}} \Vert v_{n} \Vert _{\lambda }^{r} \\ \leq & \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b \bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{ \ast }}}\bar{S}^{-2}C_{E}^{2} \biggr) ^{\frac{2_{\ast }-r}{2_{\ast }-2}} \bigl( \bar{S}^{-2_{\ast }}C_{E}^{2_{\ast }} \bigr) ^{\frac{r-2}{2_{ \ast }-2}} \\ & {} \cdot \biggl[ \biggl( \frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}- \frac{1}{ \tau } \biggr) -\frac{C_{2}}{\lambda b} \biggr) ^{-1}C_{3} \biggr] ^{ \frac{r}{2}}+o(1). \end{aligned}$$
(3.18)

Therefore, from (2.2), (3.1), (3.17), and (3.18), we have

$$\begin{aligned} o(1) =& \int_{\mathbb{R}^{N}}\bigl[ \vert \Delta v_{n} \vert ^{2}+ \vert \nabla v_{n} \vert ^{2}+V _{\lambda }(x)v_{n}^{2}\bigr] \,dx- \int_{\mathbb{R}^{N}}f(x,v_{n})v_{n} \,dx \\ \geq &\frac{\mu_{0}-1}{\mu_{0}} \Vert v_{n} \Vert _{\lambda }^{2}- \varepsilon \int_{\mathbb{R}^{N}} \vert v_{n} \vert ^{2} \,dx-C(\varepsilon ) \int_{\mathbb{R} ^{N}} \vert v_{n} \vert ^{p}\,dx \\ \geq & \biggl( \frac{\mu_{0}-1}{\mu_{0}}-\frac{\varepsilon }{\lambda b} \biggr) \Vert v_{n} \Vert _{\lambda }^{2} -C(\varepsilon ) \biggl( \int_{\mathbb{R}^{N}} \vert v _{n} \vert ^{p}\,dx \biggr) ^{\frac{p-2}{p}} \biggl( \int_{\mathbb{R}^{N}} \vert v_{n} \vert ^{p}\,dx \biggr) ^{\frac{2}{p}} \\ \geq & \biggl( \frac{\mu_{0}-1}{\mu_{0}}-\frac{\varepsilon }{\lambda b} \biggr) \Vert v_{n} \Vert _{\lambda }^{2} \\ & {} -C(\varepsilon ) \biggl[ \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{\ast }}}\bar{S}^{-2}C _{E}^{2} \biggr) ^{\frac{2_{\ast }-r}{2_{\ast }-2}} \bigl( \bar{S}^{-2_{ \ast }}C_{E}^{2_{\ast }} \bigr) ^{\frac{r-2}{2_{\ast }-2}} \biggr] ^{ \frac{p-2}{p}} \\ \geq & \Vert v_{n} \Vert _{\lambda }^{2} \biggl\{ \biggl( \frac{\mu_{0}-1}{\mu _{0}}-\frac{\varepsilon }{\lambda b} \biggr) \\ & {}-C(\varepsilon ) \biggl[ \biggl( \frac{1}{ \lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{\ast }}}\bar{S} ^{-2}C_{E}^{2} \biggr) ^{\frac{2_{\ast }-r}{2_{\ast }-2}} \bigl( \bar{S} ^{-2_{\ast }}C_{E}^{2_{\ast }} \bigr) ^{\frac{r-2}{2_{\ast }-2}} \biggr] ^{\frac{p-2}{p}} \\ & {} \cdot \biggl[ \biggl( \frac{\mu_{0}-1}{\mu_{0}} \biggl( \frac{1}{2}- \frac{1}{ \tau } \biggr) -\frac{C_{2}}{\lambda b} \biggr) ^{-1}C_{3} \biggr] ^{ \frac{p-2}{2}} \biggl[ \biggl( \frac{1}{\lambda b} \biggr) ^{\frac{2_{\ast }-r}{2_{ \ast }-2}} \bigl( \bar{S}^{-2_{\ast }}C_{E}^{2_{\ast }} \bigr) ^{\frac{p-2}{2_{ \ast }-2}} \biggr] ^{\frac{2}{p}} \biggr\} +o(1). \end{aligned}$$

Therefore, there exists \(\Lambda =\Lambda (C_{3})\geq \Lambda_{0}\) such that \(v_{n}\rightarrow 0\) in \(E_{\lambda }\) for \(\lambda >\Lambda \). This completes the proof. □

Proofs of the main results

In this section, we devote ourselves to giving the proofs of Theorems 1.11.3.

Proof of Theorem 1.1

Lemma 3.1 shows that the functional \(I_{\lambda }\) satisfies the geometric property of the mountain pass theorem. Moreover, Lemmas 3.3 and 3.4 imply that \(I_{\lambda }\) satisfies the \((C)_{c}\) condition for any \(c\in \mathbb{R}\). Then Theorem 1.1 follows from Proposition 2.1. This completes the proof. □

Proof of Theorem 1.2

Noting that \(p>2\) for \(\lambda >\Lambda \) (where Λ is defined in Lemma 3.4), ε and \(\Vert u \Vert _{\lambda }\) sufficiently small, one has

$$\begin{aligned} \begin{aligned}[b] & \biggl[ \frac{\mu_{0}-1}{2\mu_{0}}-\varepsilon \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{\ast }}}\bar{S}^{-2}C _{E}^{2} \biggr) \biggr] \Vert u \Vert _{\lambda }^{2} \\ &\quad \geq 2C(\varepsilon ) \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{ \ast }-2}{2_{\ast }}}\bar{S}^{-2}C_{E}^{2} \biggr) ^{\frac{2_{\ast }-p}{2_{ \ast }-2}} \bigl( \bar{S}^{-2_{\ast }}C_{E}^{2_{\ast }} \bigr) ^{\frac{p-2}{2_{ \ast }-2}} \Vert u \Vert _{\lambda }^{p}. \end{aligned} \end{aligned}$$
(4.1)

Set

$$ B_{\rho }:=\bigl\{ u\in E_{\lambda }: \Vert u \Vert _{\lambda }< \rho \bigr\} . $$

Then, for any \(u\in \overline{B}_{\rho }\), ε and ρ sufficiently small, it follows from (2.1), (2.3), (2.4), (3.2), and (4.1) that

$$\begin{aligned} I_{\lambda }(u) \geq &\frac{\mu_{0}-1}{2\mu_{0}} \Vert u \Vert _{\lambda }^{2} - \int_{\mathbb{R}^{N}} F(x,u)\,dx \\ \geq &\frac{\mu_{0}-1}{2\mu_{0}} \Vert u \Vert _{\lambda }^{2} - \int_{\mathbb{R}^{N}} \bigl[\varepsilon \vert u \vert ^{2}+C( \varepsilon ) \vert u \vert ^{p}\bigr]\,dx \\ \geq & \biggl[ \frac{\mu_{0}-1}{2\mu_{0}}-\varepsilon \biggl( \frac{1}{ \lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{\ast }}} \bar{S} ^{-2}C_{E}^{2} \biggr) \biggr] \Vert u \Vert _{\lambda }^{2} \\ & {} -C(\varepsilon ) \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{ \ast }-2}{2_{\ast }}}\bar{S}^{-2}C_{E}^{2} \biggr) ^{\frac{2_{\ast }-p}{2_{ \ast }-2}} \bigl( \bar{S}^{-2_{\ast }}C_{E}^{2_{\ast }} \bigr) ^{\frac{p-2}{2_{ \ast }-2}} \Vert u \Vert _{\lambda }^{p} \\ \geq &\frac{1}{2} \biggl[ \frac{\mu_{0}-1}{2\mu_{0}}-\varepsilon \biggl( \frac{1}{ \lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{\ast }-2}{2_{\ast }}}\bar{S} ^{-2}C_{E}^{2} \biggr) \biggr] \Vert u \Vert _{\lambda }^{2}. \end{aligned}$$

Hence,

$$ I_{\lambda }| _{\partial B_{\rho }}\geq \frac{1}{2} \biggl[ \frac{\mu_{0}-1}{2 \mu_{0}}-\varepsilon \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{ \ast }-2}{2_{\ast }}}\bar{S}^{-2}C_{E}^{2} \biggr) \biggr] \rho^{2}:= \alpha >0. $$

Since \(E_{\lambda }\) is a separable Hilbert space, \(E_{\lambda }\) has a countable orthogonal basis \(\{e_{j}\}\). Let \(E_{\lambda }^{k}:= \operatorname{span}\{e_{1},\ldots,e_{k}\}\) and \(Z_{\lambda }^{k}=(E_{\lambda }^{k})^{\bot }\). Then \(E_{\lambda }=E_{\lambda }^{k}\oplus Z_{ \lambda }^{k}\). Therefore, for ε and ρ sufficiently small, we obtain

$$ I_{\lambda }| _{\partial B_{\rho }\cap Z_{\lambda }^{k}}\geq \alpha >0. $$

Moreover, for any finite dimensional subspace \(\overline{E}\subset E _{\lambda }\), there is a positive integral number m such that \(\overline{E}\subset E_{\lambda }^{m}\). Note that all norms are equivalent in a finite dimensional space, then a direct calculation from (2.4) and (3.6) gives that

$$\begin{aligned} I_{\lambda }(u) =&\frac{1}{2} \Vert u \Vert _{\lambda }^{2}- \frac{1}{2} \int_{\mathbb{R}^{N}}V^{-}(x)u^{2}\,dx - \int_{\mathbb{R}^{N}} F(x,u)\,dx \\ \leq &\frac{1}{2} \Vert u \Vert _{\lambda }^{2} - \int_{\mathbb{R}^{N}} \bigl\vert F(x,u) \bigr\vert \,dx \\ \leq &\frac{1}{2} \Vert u \Vert _{\lambda }^{2}-M \Vert u \Vert _{\lambda }^{q}+\bar{M} \Vert u \Vert _{\lambda }^{2}. \\ \leq &C_{4} \Vert u \Vert _{\lambda }^{2}-C_{5} \Vert u \Vert _{\lambda }^{q}. \end{aligned}$$

Consequently, there is large \(\gamma >0\) such that \(I_{\lambda }(u) \leq 0\) on \(\overline{E}\setminus B_{\gamma }\). Therefore, there is a point \(e\in E_{\lambda }\) with \(\Vert e \Vert _{\lambda }>\rho \) such that \(I_{\lambda }(e)<0\).

Obviously, \(I_{\lambda }(0)=0\) and condition \((F5)\) implies that the functional \(I_{\lambda }\) is even. Therefore, combining the arguments above with Lemmas 3.3 and 3.4, Proposition 2.2 implies that the functional \(I_{\lambda }\) possesses an unbounded sequence of critical values, that is, problem (1.1) has infinitely many high energy solutions. □

Proof of Theorem 1.3

Following the argument in [1] (or see [31]), for any sequence \(\lambda_{n}\rightarrow \infty \), we let \(u_{n}:=u_{\lambda_{n}}\) be the critical points of \(I_{\lambda }\) obtained in Theorem 1.1. By similar arguments of Lemma 3.3, we get that \(\Vert u_{n} \Vert _{\lambda_{n}}\) is bounded in \(E_{\lambda }\), that is,

$$ \Vert u_{n} \Vert _{\lambda_{n}}\leq C_{6}, $$
(4.2)

where \(C_{6}\) is independent of \(\lambda_{n}\). Therefore, we may assume that \(u_{n}\rightharpoonup u_{0}\) in E and \(u_{n}\rightarrow u_{0}\) in \(L_{\mathrm {loc}}^{s}(\mathbb{R}^{N})\) for \(2\leq s< 2_{\ast }\). Then Fatou’s lemma implies that

$$ \int_{\mathbb{R}^{N}}V^{+}(x)u_{0}^{2}\,dx \leq \liminf_{n\rightarrow \infty } \int_{\mathbb{R}^{N}}V^{+}(x)u_{n}^{2}\,dx \leq \liminf_{n\rightarrow \infty }\frac{ \Vert u_{n} \Vert _{\lambda_{n}}^{2}}{ \lambda_{n}}=0, $$

which implies that \(u_{0}=0\) a.e. in \(\mathbb{R}^{N}\setminus V^{-1}(0)\) and \(u_{0}\in H_{0}^{2}(\Omega )\) by \((V3)\). For any \(\varphi \in C _{0}^{\infty }(\Omega )\), it follows from \(\langle I'_{\lambda }(u _{n}),\varphi \rangle =0\) that

$$ \int_{\mathbb{R}^{N}}\bigl[\triangle u_{0}\triangle \varphi + \nabla u_{0} \nabla \varphi - V^{-}(x)u_{0}\varphi \bigr] \,dx= \int_{\mathbb{R}^{N}}f(x,u _{0})\varphi \,dx, $$

which means that \(u_{0}\) is a weak solution of problem (1.2) by the density of \(C_{0}^{\infty }(\Omega )\) in \(H_{0}^{2}(\Omega )\).

Now, we show that \(u_{n}\rightarrow u_{0}\) in \(L^{s}(\mathbb{R}^{N})\) for \(2\leq s<2_{\ast }\). If not, by Lions’ vanishing lemma [12], there exist \(\delta_{0}>0\), \(R_{0}>0\), and \(x_{n}\in \mathbb{R}^{N}\) such that

$$ \int_{B_{R_{0}}(x_{n})}(u_{n}-u_{0})^{2}\,dx \geq \delta_{0}. $$

Moreover, \(x_{n}\rightarrow \infty \), hence \(\vert B_{R_{0}}(x_{n})\cap \{x \in \mathbb{R}^{N}\mid V^{+}(x)< b\} \vert \rightarrow 0\). By the Hölder inequality, we have

$$ \int_{B_{R_{0}}(x_{n})\cap \{x\in \mathbb{R}^{N}\mid V^{+}(x)< b\}}(u_{n}-u _{0})^{2} \,dx\rightarrow 0. $$

Consequently,

$$\begin{aligned} \Vert u_{n} \Vert _{\lambda }^{2} \geq & \lambda_{n}b \int_{B_{R_{0}}(x_{n})\cap \{x\in \mathbb{R}^{N}\mid V^{+}(x)\geq b\}}u _{n}^{2}\,dx \\ \geq &\lambda_{n}b \int_{B_{R_{0}}(x_{n})\cap \{x\in \mathbb{R}^{N}\mid V^{+}(x)\geq b\}}(u _{n}-u_{0})^{2} \,dx \\ =& \lambda_{n}b \biggl( \int_{B_{R_{0}}(x_{n})}(u_{n}-u_{0})^{2} \,dx- \int_{B_{R_{0}}(x_{n})\cap \{x\in \mathbb{R}^{N}\mid V^{+}(x)< b\}}(u_{n}-u _{0})^{2} \,dx+o(1) \biggr) \\ \rightarrow &\infty , \end{aligned}$$

which contradicts (4.2). Therefore, \(u_{n}\rightarrow u_{0}\) in \(L^{s}(\mathbb{R}^{N})\) for \(2\leq s<2_{\ast }\). Furthermore, applying \((F1)\), \((F2)\) and \(u_{n}\rightarrow u_{0}\) in \(L^{s}(\mathbb{R}^{N})\) gives that

$$ \int_{\mathbb{R}^{N}} f(x,u_{n})u_{n}\,dx\rightarrow \int_{\mathbb{R} ^{N}} f(x,u_{0})u_{0}\,dx. $$
(4.3)

For \(\varepsilon \in ( 0,\frac{\mu_{0}-1}{\mu_{0}} ( \frac{1}{ \lambda b} + \vert \{V^{+}(x)< b\} \vert ^{\frac{2_{\ast }-2}{2_{\ast }}}\bar{S} ^{-2}C_{E}^{2} ) ^{-1} ) \), (3.1) implies that

$$ \biggl\vert \int_{\mathbb{R}^{N}} f(x,u_{n})u_{n}\,dx \biggr\vert \leq \int_{\mathbb{R}^{N}} \bigl[\varepsilon \vert u_{n} \vert ^{2}+C(\varepsilon ) \vert u_{n} \vert ^{p} \bigr]\,dx, $$
(4.4)

which combining with the facts \(u_{n}\neq 0\), (2.1), (2.3), and (4.3) yields that

$$\begin{aligned} \frac{\mu_{0}-1}{\mu_{0}} \Vert u_{n} \Vert _{\lambda_{n}}^{2} \leq & \int_{\mathbb{R}^{N}}\bigl[ \vert \Delta u_{n} \vert ^{2}+ \vert \nabla u_{n} \vert ^{2}+V_{\lambda _{n}}(x)u_{n}^{2} \bigr]\,dx= \int_{\mathbb{R}^{N}}f(x,u_{n})u_{n}\,dx \\ \leq & \int_{\mathbb{R}^{N}} \bigl[\varepsilon \vert u_{n} \vert ^{2}+C(\varepsilon ) \vert u _{n} \vert ^{p} \bigr]\,dx \\ \leq & \varepsilon \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{ \ast }-2}{2_{\ast }}}\bar{S}^{-2}C_{E}^{2} \biggr) \Vert u_{n} \Vert _{\lambda _{n}}^{2} \\ & {} +C(\varepsilon ) \biggl( \frac{1}{\lambda b} + \bigl\vert \bigl\{ V^{+}(x)< b\bigr\} \bigr\vert ^{\frac{2_{ \ast }-2}{2_{\ast }}}\bar{S}^{-2}C_{E}^{2} \biggr) ^{\frac{2_{\ast }-p}{2_{ \ast }-2}} \bigl( \bar{S}^{-2_{\ast }}C_{E}^{2_{\ast }} \bigr) ^{\frac{p-2}{2_{ \ast }-2}} \Vert u \Vert _{\lambda_{n}}^{p}, \end{aligned}$$

which implies that

$$ \Vert u_{n} \Vert _{\lambda_{n}}\geq \biggl[ \frac{\frac{\mu_{0}-1}{\mu_{0}}- \varepsilon ( \frac{1}{\lambda b} + \vert \{V^{+}(x)< b\} \vert ^{\frac{2_{ \ast }-2}{2_{\ast }}}\bar{S}^{-2}C_{E}^{2} ) }{C(\varepsilon ) ( \frac{1}{\lambda b} + \vert \{V^{+}(x)< b\} \vert ^{\frac{2_{\ast }-2}{2_{ \ast }}}\bar{S}^{-2}C_{E}^{2} ) ^{\frac{2_{\ast }-p}{2_{\ast }-2}} ( \bar{S}^{-2_{\ast }}C_{E}^{2_{\ast }} ) ^{\frac{p-2}{2_{ \ast }-2}}} \biggr] ^{\frac{1}{p-2}}>0. $$
(4.5)

Moreover, it follows from \(\langle I'_{\lambda_{n}}(u_{n}),u_{n} \rangle =0\) that

$$ \int_{\mathbb{R}^{N}}f(x,u_{n})u_{n}\,dx= \int_{\mathbb{R}^{N}}\bigl[ \vert \Delta u_{n} \vert ^{2}+ \vert \nabla u_{n} \vert ^{2}+V_{\lambda_{n}}(x)u_{n}^{2} \bigr]\,dx \geq \frac{ \mu_{0}-1}{\mu_{0}} \Vert u_{n} \Vert _{\lambda_{n}}^{2}. $$
(4.6)

Then, a direct calculation from (4.4), (4.5), and (4.6) shows that

$$\begin{aligned} \int_{\mathbb{R}^{N}} f(x,u_{0})u_{0}\,dx&\geq \frac{\mu_{0}-1}{\mu_{0}} \biggl[ \frac{\frac{\mu_{0}-1}{\mu_{0}}-\varepsilon ( \frac{1}{ \lambda b} + \vert \{V^{+}(x)< b\} \vert ^{\frac{2_{\ast }-2}{2_{\ast }}}\bar{S} ^{-2}C_{E}^{2} ) }{C(\varepsilon ) ( \frac{1}{\lambda b} + \vert \{V ^{+}(x)< b\} \vert ^{\frac{2_{\ast }-2}{2_{\ast }}}\bar{S}^{-2}C_{E}^{2} ) ^{\frac{2_{\ast }-p}{2_{\ast }-2}} ( \bar{S}^{-2_{\ast }}C_{E} ^{2_{\ast }} ) ^{\frac{p-2}{2_{\ast }-2}}} \biggr] ^{\frac{2}{p-2}}\\&>0, \end{aligned}$$

which means that \(u_{0}\neq 0\).

In what follows, we shall show that \(u_{n}\rightarrow u_{0}\) in E. Since \(\langle I'_{\lambda_{n}}(u_{n}),u_{n}\rangle =\langle I'_{ \lambda_{n}}(u_{n}),u_{0}\rangle =0\), we have

$$ \Vert u_{n} \Vert _{\lambda_{n}}^{2}- \int_{\mathbb{R}^{N}}V^{-}(x)u_{n}^{2}\,dx= \int_{\mathbb{R}^{N}}f(x,u_{n})u_{n}\,dx $$
(4.7)

and

$$ \langle u_{n},u_{0}\rangle_{\lambda_{n}}- \int_{\mathbb{R}^{N}}V^{-}(x)u _{n} u_{0} \,dx= \int_{\mathbb{R}^{N}}f(x,u_{n})u_{0}\,dx. $$
(4.8)

Similar to the proof of (3.10), we have

$$\begin{aligned} \int_{\mathbb{R}^{N}}V^{-}(x) \bigl(u_{n}^{2}-u_{n}u_{0} \bigr)\,dx \leq & \biggl\vert \int _{\mathbb{R}^{N}}V^{-}(x) \bigl(u_{n}^{2}-u_{n}u_{0} \bigr)\,dx \biggr\vert \\ =& \biggl\vert \int_{\operatorname {supp}V^{-}}V^{-}(x)u_{n}(u_{n}-u_{0}) \,dx \biggr\vert \\ \leq & \bigl\Vert V^{-} \bigr\Vert _{\infty } \biggl( \int_{\operatorname {supp}V^{-}} \vert u_{n}-u_{0} \vert ^{2}\,dx \biggr) ^{\frac{1}{2}} \biggl( \int_{\mathbb{R}^{N}} \vert u_{n} \vert ^{2} \biggr) ^{ \frac{1}{2}} \\ \rightarrow &0, \end{aligned}$$
(4.9)

since \(u_{n}\rightarrow u_{0}\) in \(L_{\mathrm {loc}}^{2}(\mathbb{R}^{N})\) and \(\{u_{n}\}\) is bounded in \(L^{2}(\mathbb{R}^{N})\). Moreover, combining (4.4) with (4.7)–(4.9) shows that

$$ \lim_{n\rightarrow \infty } \Vert u_{n} \Vert _{\lambda_{n}}^{2}= \lim_{n\rightarrow \infty }\langle u_{n},u_{0}\rangle_{\lambda_{n}} = \lim _{n\rightarrow \infty }\langle u_{n},u_{0}\rangle = \Vert u_{0} \Vert ^{2}. $$

On the other hand, weak lower semi-continuity of norm implies that

$$ \lim_{n\rightarrow \infty } \Vert u_{n} \Vert _{\lambda_{n}}^{2}\geq \liminf_{n\rightarrow \infty } \Vert u_{n} \Vert ^{2}\geq \Vert u_{0} \Vert ^{2}. $$

Thus, \(u_{n}\rightarrow u_{0}\) in E as \(n\rightarrow \infty \). Therefore, \(u_{0}\) is a nontrivial solution of problem (1.2). We complete the proof. □

Conclusions

A class of biharmonic equations with sign-changing potentials and an indefinite nonlinearity is studied in the present paper. Under some suitable conditions, the existence of nontrivial solutions and the high energy solutions are obtained by using variational methods. Moreover, the phenomenon of concentration of solutions is explored. The results extend the main conclusions in recent literature.

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Acknowledgements

The authors wish to thank the reviewers and the handling editor for their comments and suggestions, which led to a great improvement in the presentation of this work.

Funding

This work is partially supported by the Scientific Research Fund of Hunan Provincial Education Department (No. 17C1362 and No. 17C1364) and the Doctoral Funds of SCU (No. 2016XQD40 and No. 2016XQD42), and the Hunan Natural Science Fund Youth Fund Project (No. 2018JJ3419).

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Xiao, Q., Liu, H. & Ouyang, Z. Existence and concentration of a nonlinear biharmonic equation with sign-changing potentials and indefinite nonlinearity. Adv Differ Equ 2018, 384 (2018). https://doi.org/10.1186/s13662-018-1782-9

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MSC

  • 35J20
  • 35J60

Keywords

  • Biharmonic equation
  • Variational methods
  • High energy solutions
  • Concentration of solutions