Skip to main content

On nontrivial solutions of nonlinear Schrödinger equations with sign-changing potential

Abstract

In this paper, we consider the superlinear Schrödinger equation with bounded potential well. The potential here is allowed to be sign-changing. Without assuming the Ambrosetti–Rabinowitz-type condition, we prove the existence of a nontrivial solution and multiplicity results.

Introduction and main results

This paper is concerned with the existence and multiplicity of nontrivial solutions for the superlinear Schrödinger equation of the form

$$ \textstyle\begin{cases} -\Delta u+V(x)u=f(x,u), \\ u\in H^{1}(\mathbb{R}^{N}),\quad N\ge 3. \end{cases} $$
(1.1)

With the aid of variational methods, problems of the form (1.1) have been extensively studied in the past decades. There are many works adopting various assumptions on V and f; see, for example, [113] and references therein.

Motivated by the above works, in this paper, we consider equation (1.1) with a sign-changing potential well. For the potential V, we assume:

  1. (V)

    \(V\in C(\mathbb{R}^{N})\), \(V(x)< V_{\infty }:=\lim_{ \vert x \vert \rightarrow \infty }V ( x )<\infty \), \(0\notin \sigma ( -\Delta +V ) \), the spectrum of \(-\Delta +V\).

Remark 1.1

Define the nondecreasing sequence of minimax values by

$$ \lambda _{n}=\lim_{S\in \mathcal{S}_{n}}\sup _{u\in S\backslash \{ \mathbf{0}\}}\frac{\int _{\mathbb{R}^{N}} ( \vert \nabla u \vert ^{2}+V(x)u^{2} ) \,\mathrm{d}x}{\int _{\mathbb{R}^{N}}u^{2}\,\mathrm{d}x}, \quad n\in \mathbb{N}, $$

where \(\mathcal{S}_{n}\) is a family of n-dimensional subspaces of \(C_{0}^{\infty }(\mathbb{R}^{N})\). We can see that \(\sigma _{\mathrm{ess}}(-\Delta +V)\in ( V_{\infty },\infty ) \) by (V), \(\lambda _{\infty }:=\lim_{k\rightarrow \infty }\lambda _{n}=\inf \sigma _{\mathrm{ess}} ( -\Delta +V ) <\infty \), and \(\lambda _{n}\in \sigma _{pp} ( -\Delta +V ) \) whenever \(\lambda _{n}<\lambda _{\infty }\), where \(\sigma _{\mathrm{ess}} ( -\Delta +V ) \) denotes the essential spectrum of \(-\Delta +V\), and \(\sigma _{pp} ( -\Delta +V ) \) denotes the pure point spectrum of \(-\Delta +V\) (see [14, 15] for details).

Besides (V), in [11, 12], it is also assumed that \(\inf V>0\), so that \(\lambda _{1}>0\). Then the energy functional with respect to problem (1.1) has mountain pass geometry. In this work, we are interested in the case where the Schrödinger operator \(-\Delta +V\) possesses a nontrivial negative space, which leads to more difficulty in verifying the compactness conditions. To the best of our knowledge, there are not many results in this case.

In this paper, we do not assume any compactness conditions on the potential function V. It is well known that the main difficulty in studying (1.1) in \(\mathbb{R}^{N}\) is the lack of compactness. This difficulty can be avoided for (1.1) in bounded domains or if the potential function V possesses some compactness conditions. For example, if \(\lim_{|x|\to \infty } V(x) = \infty \) or u is radially symmetric, we can get some compactness embedding, and then the Palias–Smale condition can be proved. We refer to [16] in this direction.

Denote \(F(x,t):=\int _{0}^{t}f(x,s)\,\mathrm{d}s\), \(2^{\ast }:=\frac{2N}{N-2}\), and \(p^{\prime }:=\frac{p}{p-1}\), the conjugate exponent of p. We make the following assumptions on the nonlinearity f.

(\(f_{1}\)):

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

$$ \bigl\vert f(x,t) \bigr\vert \le c\bigl(1+ \vert t \vert ^{p-1}\bigr) $$

for \(x\in \mathbb{R}^{N}\) and \(t\in \mathbb{R}\).

(\(f_{2}\)):

\(f(x,t)=o(t)\) as \(t\rightarrow 0\) uniformly in \(x\in \mathbb{R}^{N}\).

(\(f_{3}\)):

\(F(x,t)/t^{2}\rightarrow \infty \) as \(|t|\rightarrow \infty \) uniformly in \(x\in \mathbb{R}^{N}\).

(\(f_{4}\)):

\(\lim_{|x|\rightarrow \infty }\sup_{|t|\leq l} \frac{ \vert f ( x,t ) \vert }{|t|}=0\) for every \(l>0\).

(\(f_{5}\)):

There exist \(a,b>0\) and \(\alpha \in (0, \alpha _{\ast })\) such that

$$ 0< \biggl( 2+\frac{1}{a \vert t \vert ^{\alpha }+b} \biggr) F(x,t)\le tf(x,t) $$

for \(x\in \mathbb{R}^{N}\) and \(t\neq 0\), where \(\alpha _{\ast }=\min \{p',(2^{\ast }-1)p^{\prime }-2^{\ast }\}\).

Then we have the following two results.

Theorem 1.2

Under assumptions (V) and (\(f_{1}\))(\(f_{5}\)), problem (1.1) possesses at least one nontrivial solution.

Theorem 1.3

Under assumptions of Theorem 1.2, if \(f(x,t)\) is odd in t, then problem (1.1) possesses infinitely many solutions.

Remark 1.4

To produce critical points of the variational functional of (1.1), we will eventually encounter the compactness problem. For this issue, we introduced assumption (\(f_{4}\)). It is easy to see that if \(a:\mathbb{R}^{N}\rightarrow \mathbb{R}\) is continuous, \(\lim_{ \vert x \vert \rightarrow \infty }a(x)=0\), and \(p\in (2,2^{\ast })\), then

$$ f(x,t)=a(x) \vert t \vert ^{p-2}t $$

satisfies (\(f_{1}\))–(\(f_{5}\)).

Remark 1.5

Most papers concerned with the superlinear Schrödinger equations involve the following classical condition of Ambrosetti and Rabinowitz:

  1. (AR)

    There exists \(\mu >2\) such that \(0<\mu F(x,t)\le tf(x,t)\) for all \(x\in \mathbb{R}^{N}\) and \(t\neq 0\).

Condition (AR) plays a crucial role in proving the boundedness of Palias–Smale or Cerami sequences. Instead, we introduce a new condition (\(f_{5}\)), and we will illustrate a general technique to establish the boundedness of Cerami sequences. It is well known that many superlinear nonlinearities such as

$$ f(x,t)=t\ln \bigl(1+ \vert t \vert \bigr) $$

do not satisfy condition (AR). Note that \(\frac{1}{a|t|^{\alpha }+b}\rightarrow 0\) as \(\vert t \vert \rightarrow \infty \), which indicates that (\(f_{5}\)) is somewhat weaker than (AR). Note also that \((2^{\ast }-1)p^{\prime }-2^{\ast }>0\) whenever \(p<2^{\ast }\). So the parameter \(\alpha \in (0, \alpha _{\ast })\) is available. It is also worth pointing out that (\(f_{5}\)) is not a superlinear condition. Indeed, there are asymptotically linear functions satisfying (\(f_{5}\)).

Preliminaries

We denote by \(E:=H^{1}(\mathbb{R}^{N})\) the usual Sobolev space. Define the functional \(\Phi :E\rightarrow \mathbb{R}\) by

$$ \Phi (u)=\frac{1}{2} \int _{\mathbb{R}^{N}} \bigl( \vert \nabla u \vert ^{2}+V(x)u^{2} \bigr) \,\mathrm{d}x- \int _{\mathbb{R}^{N}}F(x,u)\,\mathrm{d}x. $$

Our assumptions on V and f stated above imply that the Schrödnger operator \(-\Delta +V\) is selfadjoint and semibounded in \(L^{2}(\mathbb{R}^{N})\) and \(\Phi \in C^{1}(E,\mathbb{R})\). A direct computation gives that, for all \(u,v\in E\),

$$ \bigl\langle \Phi ^{\prime }(u),v \bigr\rangle = \int _{\mathbb{R}^{N}} \bigl( \nabla u\nabla v+V(x)uv \bigr) \,\mathrm{d}x- \int _{\mathbb{R}^{N}}f(x,u)v \,\mathrm{d}x. $$

It is well known that the critical points of Φ are solutions of problem (1.1).

By (V) 0 is not an eigenvalue of \(-\Delta +V\). If \(\lambda _{1}>0\),we easily see that Φ has the mountain pass geometry. This case is simple, and we omit it here. In view of Remark 1.1, we arrange the eigenvalues (counted with multiplicity) of \(-\Delta +V\) as

$$ -\infty < \lambda _{1}\le \lambda _{2}\le \cdots \le \lambda _{\ell }< 0< \lambda _{\ell +1}\le \cdots < \lambda _{\infty } $$
(2.1)

and denote by \(e_{j}\) the corresponding eigenfunction of \(\lambda _{j}\). Let \(E^{-}=\operatorname{span} \{ e_{1},\ldots ,e_{\ell } \} \) and \(E^{+}= ( E^{-} ) ^{\bot }\). From (V) we deduce that \(E=E^{-}\oplus E^{+}\), where \(E^{-}\) and \(E^{+}\) are the negative and positive eigenspaces of the operator \(-\Delta +V\), and that \(\dim E^{-}<\infty \). For \(u,v\in E\), define

$$ ( u,v ) = \int _{\mathbb{R}^{N}} \bigl( \nabla u^{+} \nabla v^{+}+V(x)u^{+}v^{+} \bigr) \,\mathrm{d}x- \int _{\mathbb{R}^{N}} \bigl( \nabla u^{-}\nabla v^{-}+V(x)u^{-}v^{-} \bigr) \,\mathrm{d}x, $$
(2.2)

where \(u=u^{-}+u^{+}\) with \(u^{-}\in E^{-}\) and \(u^{+}\in E^{+}\). Then \(( \cdot ,\cdot ) \) is an inner product on E. Therefore E is a Hilbert space with the norm \(\Vert \cdot \Vert := \sqrt{ ( \cdot ,\cdot ) }\). We easily see that

$$ \Phi (u)=\frac{1}{2} \bigl\Vert u^{+} \bigr\Vert ^{2}-\frac{1}{2} \bigl\Vert u^{-} \bigr\Vert ^{2}- \int _{\mathbb{R}^{N}}F(x,u) \,\mathrm{d}x $$
(2.3)

and

$$ \bigl\langle \Phi ^{\prime }(u),v \bigr\rangle =\bigl(u^{+},v^{+} \bigr)-\bigl(u^{-},v^{-}\bigr)- \int _{\mathbb{R}^{N}}f(x,u)v\,\mathrm{d}x. $$

For any \(s\in [ 2,2^{\ast } ] \), the imbedding \(E\hookrightarrow L^{s}(\mathbb{R}^{N})\) is continuous. Consequently, there exists a constant \(\tau _{s}>0\) such that

$$ \vert u \vert _{s}\le \tau _{s} \Vert u \Vert , \quad \forall u\in E, $$
(2.4)

where \(\vert \cdot \vert _{s}\) denotes the \(L^{s}\) norm.

We next recall some abstract critical point theorems, which will be used in the proofs of our main results.

Definition 2.1

Let E be a Banach space, and let \(\Phi \in C^{1}(E,\mathbb{R})\). Given \(c\in \mathbb{R}\), a sequence \(\{ u_{n} \} \subset E\) is called a Cerami sequence of Φ at level c (shortly, a (C)c sequence) if

$$ \Phi (u_{n})\rightarrow c,\quad \bigl(1+ \Vert u_{n} \Vert \bigr) \bigl\Vert \Phi ^{ \prime }(u_{n}) \bigr\Vert \rightarrow 0. $$
(2.5)

We say that Φ satisfies the Cerami condition at level c (shortly, condition (C)c) if every (C)c sequence of Φ contains a convergent subsequence. If Φ satisfies condition (C)c for every \(c\in \mathbb{R}\), then we say that Φ satisfies the Cerami condition (shortly, condition (C)).

Obviously, condition (C) is weaker than the Palais–Smale condition. However, as was shown in [17], the deformation theorem is still valid under the Cerami condition. Thus we have the following theorems.

Theorem 2.2

(Linking theorem [18])

Let \(E=E^{-}\oplus E^{+}\) be a Banach space with \(\dim E^{-}<\infty \). Let \(R>r>0\), and let \(\phi \in E^{+}\backslash \{ 0 \} \). Define

$$ M:=\bigl\{ u=u^{-}+\lambda \phi \vert \;u^{-} \in E^{-}, \Vert u \Vert \le R, \lambda \ge 0 \bigr\} , \qquad N:=\bigl\{ u\in E^{+}\vert \Vert u \Vert =r \bigr\} . $$

If \(\Phi \in C^{1}(E,\mathbb{R})\) satisfies condition (C) and

$$ \inf_{N}\Phi >\max_{\partial M}\Phi , $$

then Φ has a nontrivial critical point.

For the proofs of Theorems 1.21.3, we will use the following fountain theorem, which is a generalization of the classical fountain theorem of Bartsch [19] (see also [10]). For \(k\in \mathbb{N}\), let

$$ Y_{k}=\operatorname{span} \{ e_{1},\ldots e_{k} \} ,\qquad Z_{k}=Y_{k}^{\perp }. $$
(2.6)

Theorem 2.3

(Fountain theorem [20])

Suppose that the functional \(\Phi \in C^{1}(E,\mathbb{R})\) is even and satisfies condition (C). Suppose that for every \(k\ge k_{0}\) for some constant \(k_{0}>0\), there exist \(\rho _{k}>r_{k}>0\) such that

(A1):

\(b_{k}=\inf_{u\in Z_{k},\Vert u\Vert =r_{k}}\Phi (u) \rightarrow \infty \) as \(k\rightarrow \infty \), and

(A2):

\(a_{k}=\max_{u\in Y_{k},\Vert u\Vert =\rho _{k}}\Phi (u)\le 0\).

Then Φ has a sequence of critical points \(\{ u_{k} \} \) such that \(\Phi (u_{k})\rightarrow \infty \).

Proof of main results

Lemma 3.1

Suppose that (V), (\(f_{1}\)), and (\(f_{2}\)) are satisfied. Then there exists \(r>0\) such that \(\inf \Phi ( {\partial B_{r}(\mathbf{0})}\cap E^{+})>0\).

Proof

It follows from (\(f_{1}\)) and (\(f_{2}\)) that, for given \(\varepsilon >0\), there is a constant \(C_{\varepsilon }>0\) such that

$$ \bigl\vert F ( x,t ) \bigr\vert \le \varepsilon \vert t \vert ^{2}+C_{\varepsilon } \vert t \vert ^{p} $$
(3.1)

and

$$ \bigl\vert f(x,t) \bigr\vert \le \varepsilon \vert t \vert +C_{\varepsilon } \vert t \vert ^{p-1}. $$
(3.2)

For \(u\in E^{+}\), we have

$$\begin{aligned} \Phi ( u ) & =\frac{1}{2} \Vert u \Vert ^{2}- \int _{\mathbb{R}^{N}}F ( x,u ) \,\mathrm{d}x \\ & \ge \frac{1}{2} \Vert u \Vert ^{2}-\varepsilon \vert u \vert _{2}^{2}-C_{\varepsilon } \vert u \vert _{p}^{p} \\ & \ge \biggl( \frac{1}{2}-\varepsilon \tau _{2} \biggr) \Vert u \Vert ^{2}-\tau _{p}C_{\varepsilon } \Vert u \Vert ^{p}, \end{aligned}$$

where \(\tau _{2}\) and \(\tau _{p}\) are constants in (2.4). Let \(\varepsilon =\frac{1}{4\tau _{2}}\). Since \(p>2\), we can fix some r small enough such that

$$ \inf_{u\in E^{+},\Vert u\Vert =r}\Phi (u)>0. $$

The proof is completed. □

Lemma 3.2

Suppose that (V) and (\(f_{1}\))(\(f_{3}\)) are satisfied. Then, for any nontrivial finite-dimensional subspace W of \(E^{+}\), there exists \(R>r\) such that \(\Phi \le 0\) in \(( E^{-}\oplus W ) \backslash B_{R}(0)\), where \(r>0\) is the constant given by Lemma 3.1.

Proof

This lemma is a corollary of [13, Lemma 2.5]. We omit the proof. □

Lemma 3.3

Suppose that (V), (\(f_{1}\))(\(f_{3}\)), and (\(f_{5}\)) are satisfied and \(c\in \mathbb{R}\). Then any (C)c sequence of Φ is bounded.

Proof

It follows from (\(f_{5}\)) that, for all \(t\neq 0\) and \(x\in \mathbb{R}^{N}\),

$$ tf(x,t)-2F(x,t)\ge \frac{1}{2a \vert t \vert ^{\alpha }+2b+1}tf(x,t)>0. $$

Let \(\{u_{n}\}\) be a (C)c sequence of Φ, that is, a sequence satisfying (2.5). Set \(\Pi _{n}:=\{x\in \mathbb{R}^{N}| |u_{n}(x)|<1\}\) and \(\Pi _{n}^{c}:=\mathbb{R}^{N}\backslash \Pi _{n}\). Then there are constants \(c_{1}\), \(c_{2}>0\) such that

$$ 2a \vert u_{n} \vert ^{\alpha }+2b+1\le 1/{c_{1}},\quad \forall x\in \Pi _{n}, $$

and

$$ 2a \vert u_{n} \vert ^{\alpha }+2b+1\le \vert u_{n} \vert ^{\alpha }/{c_{2}},\quad \forall x \in \Pi _{n}^{c}. $$

For n sufficient large, it follows that

$$\begin{aligned} D& \ge 2\Phi (u_{n})-\bigl\langle \Phi ^{\prime }(u_{n}),u_{n} \bigr\rangle \\ & = \int _{\mathbb{R}^{N}} \bigl( u_{n}f(x,u_{n})-2F(x,u_{n}) \bigr) \,\mathrm{d}x \\ & \ge \int _{\mathbb{R}^{N}} \frac{u_{n}f(x,u_{n})}{2a \vert u_{n} \vert ^{\alpha }+2b+1}\,\mathrm{d}x \\ & \ge c_{1} \int _{\Pi _{n}}u_{n}f(x,u_{n})\,\mathrm{d}x+c_{2} \int _{\Pi _{n}^{c}} \vert u_{n} \vert ^{- \alpha }u_{n}f(x,u_{n})\,\mathrm{d}x \end{aligned}$$
(3.3)

for some constant \(D>0\).

Note that \(\alpha <(2^{\ast }-1)p^{\prime }-2^{\ast }\) by (\(f_{5}\)). We have

$$ \frac{1}{p^{\prime }}< \frac{2^{\ast }}{2^{\ast }-1} \frac{1}{p^{\prime }}< \frac{2^{\ast }}{2^{\ast }+\alpha }\quad \text{and}\quad \frac{2}{2+\alpha }< \frac{2^{\ast }}{2^{\ast }+\alpha }. $$

Then we can choose a constant \(r\in (0,1)\) such that

$$ \max \biggl\{ \frac{2^{\ast }}{2^{\ast }-1}\frac{1}{p^{\prime }},\; \frac{2}{2+\alpha } \biggr\} < r< \frac{2^{\ast }}{2^{\ast }+\alpha }. $$
(3.4)

Let \(s:=r/(1-r)>0\). Then \(\frac{1}{r}+\frac{1}{-s}=1\). By (3.3) and the inverse Hölder inequality we have

$$\begin{aligned} D& \ge c_{1} \int _{\Pi _{n}}u_{n}f(x,u_{n})\,\mathrm{d}x+c_{2} \biggl( \int _{\Pi _{n}^{c}} \bigl( u_{n}f(x,u_{n}) \bigr) ^{r}\,\mathrm{d}x \biggr) ^{1/r} \biggl( \int _{\Pi _{n}^{c}} \vert u_{n} \vert ^{\alpha s}\,\mathrm{d}x \biggr) ^{1/(-s)} \\ & \ge c_{1} \int _{\Pi _{n}}u_{n}f(x,u_{n})\,\mathrm{d}x+c_{2} \frac{ ( \int _{\Pi _{n}^{c}} ( u_{n}f(x,u_{n}) ) ^{r}\,\mathrm{d}x ) ^{1/r}}{ \vert u_{n} \vert _{\alpha s}^{\alpha }}. \end{aligned}$$
(3.5)

By (\(f_{1}\)) and (\(f_{2}\)) we have

$$\begin{aligned} & \bigl\vert f(x,u) \bigr\vert ^{p^{\prime }r}\le \bigl( c_{3} \vert u \vert ^{(p-1)(p^{\prime }-1)} \bigl\vert f(x,u) \bigr\vert \bigr) ^{r}=c_{4} \bigl( uf(x,u) \bigr) ^{r}, \quad \forall \vert u \vert \ge 1, \\ & \bigl\vert f(x,u) \bigr\vert ^{2}\le c_{5} \vert u \vert \bigl\vert f(x,u) \bigr\vert =c_{5}uf(x,u), \quad \forall \vert u \vert < 1. \end{aligned}$$

Therefore by (3.5) we have

$$\begin{aligned} & \biggl( \int _{\Pi _{n}^{c}} \bigl\vert f(x,u_{n}) \bigr\vert ^{p^{\prime }r}\,\mathrm{d}x \biggr) ^{1/p^{\prime }r}\le c_{6} \vert u_{n} \vert _{\alpha s}^{\alpha /p^{ \prime }}, \end{aligned}$$
(3.6)
$$\begin{aligned} & \biggl( \int _{\Pi _{n}} \bigl\vert f(x,u_{n}) \bigr\vert ^{2}\,\mathrm{d}x \biggr) ^{1/2}\le c_{7}. \end{aligned}$$
(3.7)

In view of (3.4), we easily check that \(p^{\prime }r>1\) and \((p^{\prime }r)^{\prime },\alpha s\in {}[ 2,2^{\ast }]\), where \((p^{\prime }r)^{\prime }:=p^{\prime }r/(p^{\prime }r-1)\). Consequently, it follows from (3.6) and (3.7), Hölder’s inequality, and Sobolev’s inequality that, for n large enough and some constants \(c_{8},c_{9}>0\),

$$\begin{aligned} \bigl\Vert u_{n}^{+} \bigr\Vert ^{2}& =\bigl\langle \Phi ^{\prime }(u_{n}),~u_{n}^{+} \bigr\rangle + \int _{\mathbb{R}^{N}}f(x,u_{n})u_{n}^{+}\,\mathrm{d}x \\ & \le \bigl\Vert u_{n}^{+} \bigr\Vert + \biggl( \int _{\Pi _{n}} \bigl\vert f(x,u_{n}) \bigr\vert ^{2}\,\mathrm{d}x \biggr) ^{1/2} \bigl\vert u_{n}^{+} \bigr\vert _{2}+ \biggl( \int _{\Pi _{n}^{c}} \bigl\vert f(x,u_{n}) \bigr\vert ^{p^{ \prime }r}\,\mathrm{d}x \biggr) ^{1/p^{\prime }r} \bigl\vert u_{n}^{+} \bigr\vert _{(p^{\prime }r)^{ \prime }} \\ & \le \bigl\Vert u_{n}^{+} \bigr\Vert +c_{7} \bigl\vert u_{n}^{+} \bigr\vert _{2}+c_{6} \vert u_{n} \vert _{ \alpha s}^{\alpha /p^{\prime }} \bigl\vert u_{n}^{+} \bigr\vert _{(p^{\prime }r)^{\prime }} \le c_{8} \bigl\Vert u_{n}^{+} \bigr\Vert +c_{9} \bigl\Vert u_{n}^{+} \bigr\Vert \Vert u_{n} \Vert ^{\alpha /p^{\prime }}. \end{aligned}$$

Therefore we obtain

$$ \bigl\Vert u_{n}^{+} \bigr\Vert \le c_{8}+c_{9} \Vert u_{n} \Vert ^{\alpha /p^{ \prime }} $$

and, similarly,

$$ \bigl\Vert u_{n}^{-} \bigr\Vert \le c_{8}+c_{9} \Vert u_{n} \Vert ^{\alpha /p^{ \prime }}. $$

Note that \(\alpha < p^{\prime }\). Then we easily verify that \(\Vert u_{n}\Vert ^{2}=\Vert u_{n}^{-}\Vert ^{2}+\Vert u_{n}^{+} \Vert ^{2}\) is bounded. □

Lemma 3.4

Suppose that (V) and (\(f_{1}\))(\(f_{4}\)) are satisfied. Then any bounded (C)c sequence of Φ contains a convergent subsequence.

Proof

Suppose \(\{ u_{n} \} \) is a bounded (C)c sequence of Φ. Then, passing to a subsequence, we may assume that \(u_{n}\rightharpoonup u\) in E. Since \(\dim E^{-}<\infty \), we have \(u_{n}^{+}\rightharpoonup u^{+}\) in \(E^{+}\), \(u_{n}^{-}\rightarrow u^{-}\) in \(E^{-}\), and \(u_{n}^{+}\rightarrow u^{+}\) in \(L_{\text{loc}}^{s}(\mathbb{R}^{N})\), \(s\in {}[ 2,2^{\ast })\). To establish the strong convergence, it suffices to prove that

$$ \bigl\Vert u_{n}^{+} \bigr\Vert \rightarrow \bigl\Vert u^{+} \bigr\Vert . $$
(3.8)

Since

$$ \bigl\langle \Phi ^{\prime }(u_{n}),u_{n}^{+}-u^{+} \bigr\rangle = \bigl( u_{n}^{+},u_{n}^{+}-u^{+} \bigr) - \int _{\mathbb{R}^{N}}f(x,u_{n}) \bigl(u_{n}^{+}-u^{+} \bigr)\,\mathrm{d}x\rightarrow 0, $$

we have

$$\begin{aligned} 0 & \le \limsup_{n\rightarrow \infty } \bigl( \bigl\Vert u_{n}^{+} \bigr\Vert ^{2}- \bigl\Vert u^{+} \bigr\Vert ^{2} \bigr) \\ & =\limsup_{n\rightarrow \infty } \bigl( u_{n}^{+},u_{n}^{+}-u^{+} \bigr) =\limsup_{n\rightarrow \infty } \int _{\mathbb{R}^{N}}f(x,u_{n}) \bigl(u_{n}^{+}-u^{+} \bigr)\,\mathrm{d}x. \end{aligned}$$
(3.9)

Next, let \(\varepsilon >0\). For \(l\ge 1\), from (\(f_{1}\)) and Hölder’s inequality it follows that

$$\begin{aligned} \int _{ \vert u_{n} \vert \ge l}f(x,u_{n}) \bigl(u_{n}^{+}-u^{+} \bigr)\,\mathrm{d}x & \le 2c \int _{ \vert u_{n} \vert \ge l} \vert u_{n} \vert ^{p-1} \bigl\vert u_{n}^{+}-u^{+} \bigr\vert \,\mathrm{d}x \\ & \le 2cl^{p-2^{\ast }} \int _{ \vert u_{n} \vert \ge l} \vert u_{n} \vert ^{2^{\ast }-1} \bigl\vert u_{n}^{+}-u^{+} \bigr\vert \,\mathrm{d}x \\ & \le 2cl^{p-2^{\ast }} \vert u_{n} \vert _{2^{\ast }}^{2^{ \ast }-1} \bigl\vert u_{n}^{+}-u^{+} \bigr\vert _{2^{\ast }}. \end{aligned}$$

Since \(p<2^{\ast }\), we may fix l large enough such that

$$ \int _{ \vert u_{n} \vert \ge l}f(x,u_{n}) \bigl(u_{n}^{+}-u^{+} \bigr)\,\mathrm{d}x\le \frac{\varepsilon }{3} $$
(3.10)

for all n. Moreover, by (\(f_{4}\)) there exists \(L>0\) such that

$$ \int _{ \substack{ \vert x \vert \ge L \\ \vert u_{n} \vert \le l}}f(x,u_{n}) \bigl(u_{n}^{+}-u^{+} \bigr) \,\mathrm{d}x\le \vert u_{n} \vert _{2} \bigl\vert u_{n}^{+}-u^{+} \bigr\vert _{2}\sup_{ \vert t \vert \le l, \vert x \vert \ge L} \frac{ \vert f(x,t) \vert }{ \vert t \vert } \le \frac{\varepsilon }{3} $$
(3.11)

for all n. Finally, since \(u_{n}^{+}\rightarrow u^{+}\) in \(L^{s}(B_{L}(\mathbf{0}))\) for \(s\in {}[ 2,2^{\ast })\), from (3.2) it follows that

$$\begin{aligned} \int _{ \substack{ \vert x \vert \le L \\ \vert u_{n} \vert \le l}}f(x,u_{n}) \bigl(u_{n}^{+}-u^{+} \bigr) \,\mathrm{d}x & \le \int _{ \substack{ \vert x \vert \le L \\ \vert u_{n} \vert \le l}} \vert u_{n} \vert \bigl\vert u_{n}^{+}-u^{+} \bigr\vert \,\mathrm{d}x+C_{1} \int _{ \substack{ \vert x \vert \le L \\ \vert u_{n} \vert \le l}} \vert u_{n} \vert ^{p-1} \bigl\vert u_{n}^{+}-u^{+} \bigr\vert \,\mathrm{d}x \\ & \le \vert u_{n} \vert _{2} \bigl\vert u_{n}^{+}-u^{+} \bigr\vert _{L^{2}(B_{L}(\mathbf{0}))}+C_{1} \vert u_{n} \vert _{p}^{p-1} \bigl\vert u_{n}^{+}-u^{+} \bigr\vert _{L^{p}(B_{L}( \mathbf{0}))} \\ & \le \frac{\varepsilon }{3} \end{aligned}$$
(3.12)

for n large enough. Combining (3.10)–(3.12), we conclude that

$$ \int _{\mathbb{R}^{N}}f(x,u_{n}) \bigl(u_{n}^{+}-u^{+} \bigr)\,\mathrm{d}x\le \varepsilon $$

for n large enough. Since ε is arbitrary, this, together with (3.9), implies (3.8). The lemma is proved. □

Proof of Theorem 1.2

For \(u\in E^{-}\), since \(F(x,t)\ge 0\) by (\(f_{5}\)), we obtain that

$$ \Phi (u)=-\frac{1}{2} \Vert u \Vert ^{2}- \int _{ \mathbb{R}^{N}}F ( x,u ) \,\mathrm{d}x\le 0. $$

This, together with Lemmas 3.1 and 3.2, implies that there exist \(R>r>0\) such that

$$ \inf_{N}\Phi >0\ge \max_{\partial M}\Phi . $$

In view of Lemmas 3.3 and 3.4, Φ satisfies condition (C). By Theorem 2.2 we have that Φ possesses at least one nontrivial critical point, which is the nontrivial solution of problem (1.1). □

Proof of Theorem 1.3

Since f is odd, Φ is an even functional. By Lemmas 3.3 and 3.4 we know that Φ satisfies condition (C). To apply Theorem 2.3, it suffices to verify (A1) and (A2).

Define \(Y_{k}\) and \(Z_{k}\) as in (2.6). Recall that \(\lambda _{\ell }<0<\lambda _{\ell +1}\). If \(k>\ell \), then we have \(Z_{k}\subset E^{+}\). Define \(\beta _{k}:=\sup_{ \substack{ u\in Z_{k} \\ \Vert u \Vert =1 }} \vert u \vert _{p}\). Therefore by (2.4) and (3.1) with \(\varepsilon =1/4\tau _{2}^{2}\) we have

$$\begin{aligned} \Phi (u) & \ge \frac{1}{2} \Vert u \Vert ^{2}- \frac{1}{4\tau _{2}^{2}} \vert u \vert _{2}^{2}-C \vert u \vert _{p}^{p} \\ & \ge \frac{1}{2} \Vert u \Vert ^{2}- \frac{1}{4\tau _{2}^{2}} \vert u \vert _{2}^{2}-C \beta _{k}^{p} \Vert u \Vert ^{p} \ge \frac{1}{4} \Vert u \Vert ^{2}-C\beta _{k}^{p} \Vert u \Vert ^{p}. \end{aligned}$$

Let \(r_{k}= ( 2pC\beta _{k}^{p} ) ^{1/(2-p)}\). Then for \(u\in Z_{k}\) with \(\Vert u \Vert =r_{k}\), we have

$$ \Phi (u)\ge \frac{1}{2} \biggl( \frac{1}{2}- \frac{1}{p} \biggr) \bigl( 2pC\beta _{k}^{p} \bigr) ^{1/(2-p)}. $$

Since \(\beta _{k}\rightarrow 0\) as \(k\rightarrow \infty \) by [10, Lemma 3.8] and \(p>2\), it follows that

$$ b_{k}=\inf_{u\in Z_{k},\Vert u\Vert =r_{k}}\Phi (u) \rightarrow \infty . $$

Hence (A1) is satisfied. Finally, by Lemma 3.2 with \(W=\bigoplus_{j=0}^{k}\mathbb{R}e_{j}\) we easily see that (A2) holds. □

Availability of data and materials

Not applicable.

References

  1. 1.

    Kryszewski, W., Szulkin, A.: Generalized linking theorem with an application to a semilinear Schrödinger equation. Adv. Differ. Equ. 3(3), 441–472 (1998)

    MATH  Google Scholar 

  2. 2.

    Bartsch, T., Liu, Z., Weth, T.: Sign changing solutions of superlinear Schrödinger equations. Commun. Partial Differ. Equ. 29(1–2), 25–42 (2004). https://doi.org/10.1081/PDE-120028842

    Article  MATH  Google Scholar 

  3. 3.

    Liu, S.: On superlinear Schrödinger equations with periodic potential. Calc. Var. Partial Differ. Equ. 45(1–2), 1–9 (2012). https://doi.org/10.1007/s00526-011-0447-2

    Article  MATH  Google Scholar 

  4. 4.

    Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on \(\mathbf{R}^{N}\). Proc. R. Soc. Edinb., Sect. A 129(4), 787–809 (1999). https://doi.org/10.1017/S0308210500013147

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Coti Zelati, V., Rabinowitz, P.H.: Homoclinic type solutions for a semilinear elliptic PDE on \({\bf R}^{n}\). Commun. Pure Appl. Math. 45(10), 1217–1269 (1992). https://doi.org/10.1002/cpa.3160451002

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Zeng, J., Li, Y.: Existence of solutions for an elliptic equation with indefinite weight. Nonlinear Anal. 66(11), 2512–2519 (2007). https://doi.org/10.1016/j.na.2006.03.034

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Willem, M., Zou, W.: On a Schrödinger equation with periodic potential and spectrum point zero. Indiana Univ. Math. J. 52(1), 109–132 (2003). https://doi.org/10.1512/iumj.2003.52.2273

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Bartsch, T., Ding, Y.: On a nonlinear Schrödinger equation with periodic potential. Math. Ann. 313(1), 15–37 (1999). https://doi.org/10.1007/s002080050248

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Ding, Y., Lee, C.: Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Differ. Equ. 222(1), 137–163 (2006). https://doi.org/10.1016/j.jde.2005.03.011

    Article  MATH  Google Scholar 

  10. 10.

    Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24, p. 162. Birkhäuser, Boston (1996). https://doi.org/10.1007/978-1-4612-4146-1

    Book  MATH  Google Scholar 

  11. 11.

    Jeanjean, L., Tanaka, K.: A positive solution for an asymptotically linear elliptic problem on \(\mathbb {R}^{N}\) autonomous at infinity. ESAIM Control Optim. Calc. Var. 7, 597–614 (2002). https://doi.org/10.1051/cocv:2002068

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Li, Y., Wang, Z.-Q., Zeng, J.: Ground states of nonlinear Schrödinger equations with potentials. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23(6), 829–837 (2006). https://doi.org/10.1016/j.anihpc.2006.01.003

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257(12), 3802–3822 (2009). https://doi.org/10.1016/j.jfa.2009.09.013

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness p. 361. Academic Press, New York (1975)

    MATH  Google Scholar 

  15. 15.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators p. 396. Academic Press, New York (1978)

    MATH  Google Scholar 

  16. 16.

    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992). https://doi.org/10.1007/BF00946631

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7(9), 981–1012 (1983). https://doi.org/10.1016/0362-546X(83)90115-3

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, vol. 65, p. 100. Am. Math. Soc., Providence (1986) Published for the Conference Board of the Mathematical Sciences, Washington, DC

    Book  Google Scholar 

  19. 19.

    Bartsch, T.: Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal. 20(10), 1205–1216 (1993). https://doi.org/10.1016/0362-546X(93)90151-H

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Liu, S.B., Li, S.J.: Infinitely many solutions for a superlinear elliptic equation. Acta Math. Sinica (Chin. Ser.) 46(4), 625–630 (2003)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Professor Shibo Liu for reviewing an early draft of this paper and giving valuable comments.

Funding

This work was supported by the National Natural Science Foundation of China (nos. 11701251, 11671185, 11771195), the Natural Science Foundation of Shandong Province (nos. ZR2017BA015, ZR2019YQ04).

Author information

Affiliations

Authors

Contributions

All authors contributed equally to this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yue Wu.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Abbreviations

Not applicable.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, W., Wu, Y. & Jhang, S. On nontrivial solutions of nonlinear Schrödinger equations with sign-changing potential. Adv Differ Equ 2021, 232 (2021). https://doi.org/10.1186/s13662-021-03390-0

Download citation

Keywords

  • Schrödinger equation
  • Superlinear
  • Potential well
  • Variational methods