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.

1 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, [1–13] 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:

• (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:

• (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 ($f_{5}$).

2 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 1.2–1.3, 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 $\Phi (u_{k})\rightarrow \infty$.

3 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. □