1 Introduction and main results

In this paper, we consider the following second-order Hamiltonian system:

$$\begin{aligned} \ddot{u}(t)-L(t)u+\nabla W(t,u)=0 , \end{aligned}$$
(1)

where \(W:\mathbb{R}\times \mathbb{R}^{N} \rightarrow \mathbb{R}\) is a \(C^{1}\)-map and \(\nabla W(t,x)\) denotes the gradient with respect to the x variable. \(L\in C(\mathbb{R},\mathbb{R}^{N^{2}})\) is a matrix-valued function. A solution \(u(t)\) of problem (1) is nontrivial homoclinic (to 0) if \(u\not \equiv 0\), \(u(t)\rightarrow 0\) as \(t\rightarrow \pm \infty \).

In the last two decades, many mathematicians have successfully used variational methods to obtain the existence and multiplicity of homoclinic orbits for problem (1) such as [1, 3–18, 20–37]. Since this problem is considered in the whole space, one of the main difficulties is the lack of compactness of embedding. In 1990, Rabinowitz [20] considered the periodic case. In his paper, \(L(t)\) and \(W(t,u)\) were assumed to be periodic in t and the author obtained the results by the mountain pass theorem. Without the periodic condition, Rabinowitz and Tanaka [21] introduced the following coercive condition on \(L(t)\).

\((L')\) :

\(L\in C(\mathbb{R},\mathbb{R}^{N^{2}})\) is a symmetric and positively definite matrix for all \(t\in \mathbb{R}\), and there exists a continuous function \(l:\mathbb{R}\rightarrow \mathbb{R}^{+}\) satisfying

$$\begin{aligned} \bigl(L(t)x,x\bigr)\geq l(t) \vert x \vert ^{2} \quad \text{with } l(t)\rightarrow \infty \text{ as } \vert t \vert \rightarrow \infty . \end{aligned}$$

Using \((L')\), many mathematicians studied problem (1) with superquadratic, subquadratic, or asymptotic quadratic nonlinearities. In this paper, we mainly consider the superquadratic growth case. First, we recall some main results.

In 1991, Rabinowitz and Tanaka [21] used the so-called Ambrosetti–Rabinowitz (\((\mathit {AR})\) for short) condition to obtain the existence of homoclinic solutions for problem (1).

\((\mathit {AR})\) :

There exists a constant \(\theta > 2\) such that

$$\begin{aligned} 0 < \theta W(t,x) \leq \bigl(\nabla W(t,x),x\bigr) \end{aligned}$$

for every \(t \in \mathbb{R}\) and \(x \in \mathbb{R}^{N}\setminus \{0\}\).

Condition \((\mathit {AR})\) is useful in guaranteeing the geometrical structure of the corresponding functional and the boundedness of Palais–Smale (\((\mathit {PS})\) for short) sequence. After then, there have been many mathematicians trying to weaken this condition. In 2009, Ding and Lee [6] introduced the following generalized superquadratic condition.

\((H_{1})\) :

There exist \(\epsilon \in (0,1)\) and \(\kappa_{1}, b_{0}>0\) such that

$$\begin{aligned} \widetilde{W}(t,x)\geq b_{0}\frac{(\nabla W(t,x),x)}{ \vert x \vert ^{2-\epsilon }} \quad \text{for all } t \in \mathbb{R} \text{ and } \vert x \vert \geq \kappa_{1}, \end{aligned}$$

where

$$\begin{aligned} \widetilde{W}(t,x)=\bigl(\nabla W(t,x),x\bigr) -2W(t,x). \end{aligned}$$

In a recent paper, \((H_{1})\) was weakened by Wu et al. [28] with the following condition.

\((H_{2})\) :

\(\frac{\widetilde{W}(t,x)}{W(t,x)} \vert x \vert ^{2}\rightarrow +\infty\) as \(\vert x \vert \rightarrow \infty \) uniformly in \(t\in \mathbb{R}\).

In 2011, Yang and Han [31] also made a contribution to the multiplicity of homoclinic solutions for problem (1) under the following monotonous condition.

\((H_{3})\) :

\(s^{-1}(\nabla W(t,sx),x)\) is an increasing function of \(s\in (0,1]\) for all \((t,x)\in \mathbb{R}\times \mathbb{R}^{N}\).

In 2013, Lv and Tang [15] showed that the following condition is weaker than \((H_{3})\).

\((H_{4})\) :

There exists \(\mu \geq 1\) such that

$$\begin{aligned} \mu \widetilde{W}(t,x)\geq \widetilde{W}(t,sx) \end{aligned}$$

for all \((t,x)\in \mathbb{R}\times \mathbb{R}^{N}\) and \(s\in [0,1]\).

Conditions \((H_{1})\)–\((H_{4})\) can be seen as the generalizations or supplements of \((\mathit {AR})\). However, each result in the above papers implies that

\((H_{5})\) :

There exists \(\kappa_{2}>0\) such that \(\sup_{t\in \mathbb{R}, \vert x \vert =\kappa_{2}}W(t,x)<+\infty \).

Obviously, \((H_{5})\) holds when \(W(t,x)\) is periodic in t. If there is no periodic assumption, \((H_{5})\) is an important requirement in many papers [1, 4–13, 16–18, 20, 21, 24, 27, 33, 35]. Recently, Yuan and Zhang [35] considered the following variable separation nonlinear term:

$$\begin{aligned} W(t,x)=a(t)G(x), \end{aligned}$$
(2)

where \(a:\mathbb{R}\rightarrow \mathbb{R}^{+}\) is continuous, \(G\in C^{1}(\mathbb{R}^{N},\mathbb{R})\). In [35], the authors considered the following conditions:

\((A)\) :

\(a(t)\rightarrow 0\) as \(\vert t \vert \rightarrow \infty \);

\((L'')\) :

\(L\in C(\mathbb{R},\mathbb{R}^{N^{2}})\) is a symmetric and positively definite matrix for all \(t\in \mathbb{R}\), and there exists a constant \(M_{0}>0\) such that

$$\begin{aligned} \bigl(L(t)x,x\bigr)\geq M_{0} \vert x \vert ^{2} \quad \text{for all } (t,x)\in \mathbb{R}\times \mathbb{R}^{N}. \end{aligned}$$

Obviously, \((L'')\) is weaker than \((L')\). In a recent paper, Wu et al. [28] considered a class of fractional Hamiltonian systems with variable separated type nonlinear terms. They introduced the following condition.

\((L''')\) :

\(\frac{a(t)}{l(t)}\rightarrow 0\) as \(\vert t \vert \rightarrow \infty \), where \(l(t)\in C(\mathbb{R},\mathbb{R})\) satisfies

$$\begin{aligned} l(t)\equiv \inf_{ \vert x \vert =1}\bigl(L(t)x,x\bigr). \end{aligned}$$

Motivated by the above papers, in this paper, we introduce the following condition.

\((L_{1})\) :

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

$$\begin{aligned} \lim_{ \vert s \vert \rightarrow +\infty }\operatorname{meas} \biggl( \biggl\{ t\in B_{r_{0}}(s): \frac{l(t)}{a(t)}\ngeq M \biggr\} \biggr) =0, \quad \forall M>0, \end{aligned}$$

where \(B_{r_{0}}(s)=\{t\in \mathbb{R}: s-r_{0}\leq t\leq s+r_{0}\}\), \(l(t)\in C(\mathbb{R},\mathbb{R})\) and \(\operatorname{meas}(\cdot )\) denotes the Lebesgue measure.

In [35], the nonlinear term \(G(x)\) is assumed to satisfy the following classical AR-type condition:

\((\mathit {AR}')\) :

There exists a constant \(\nu >2\) such that

$$\begin{aligned} 0< \nu G(x)\leq \bigl(\nabla G(x),x\bigr) \end{aligned}$$

for every \(x \in \mathbb{R}^{N}\setminus \{0\}\).

From the \((\mathit {AR}')\) condition, we can deduce that

\((\mathit {SQ})\) :

\(\widetilde{G}(x)\geq 0\) for any \(x\in \mathbb{R}^{N}\), where \(\widetilde{G}(x)=(\nabla G(x),x)-2G(x)\).

In [28], the authors considered the following weakened superquadratic growth condition.

\((H_{6})\) :

There exist \(\nu >2\) and \(b_{2}, \kappa_{3}>0\) such that

$$\begin{aligned} \bigl(\nabla G(x),x\bigr)-\nu G(x)\geq -b_{2} \vert x \vert ^{2} \quad \text{for all } \vert x \vert \geq \kappa_{3}. \end{aligned}$$

\((H_{6})\) is a generalization of condition \((\mathit {AR}')\), which is only a local condition. Instead of \((H_{6})\), the authors in [28] also considered the following superquadratic condition.

\((H_{7})\) :

\(\frac{\widetilde{G}(x)}{G(x)} \vert x \vert ^{2-r_{0}}\rightarrow + \infty \) as \(\vert x \vert \rightarrow \infty \), where \(r_{0}\) is the growth exponent of the subquadratic term at infinity with respect to x.

In this paper, we consider \(G(x)\) satisfying the following monotonous condition.

\((W_{1})\) :

There exists a constant \(\theta \geq 1\) such that \(\theta \widetilde{G}(x)\geq \widetilde{G}(sx)\) for all \(x\in \mathbb{R}^{N}\) and \(s\in [0,1]\).

Now we state our main results.

Theorem 1.1

Suppose (2), \((L'')\), \((L_{1})\), \((W_{1})\) and the following conditions hold:

\((L_{2})\) :

There exists \(A>0\) such that \(a(t)\leq A l(t)\) for all \(t\in \mathbb{R}\);

\((W_{2})\) :

\(a(t)\in C(\mathbb{R},\mathbb{R}^{+})\);

\((W_{3})\) :

\(G(0)=0\) and \(\nabla G(x)=o( \vert x \vert )\) as \(\vert x \vert \rightarrow 0\);

\((W_{4})\) :

\(G(x) / \vert x \vert ^{2} \rightarrow +\infty \) as \(\vert x \vert \rightarrow \infty \).

\((W_{5})\) :

There are constants \(\zeta > 2\) and \(d_{1}>0\) such that

$$\begin{aligned} \bigl\vert G(x) \bigr\vert \leq d_{1}\bigl( \vert x \vert ^{2}+ \vert x \vert ^{\zeta }\bigr) \quad \textit{for all } x \in \mathbb{R}^{N}. \end{aligned}$$

Then problem (1) possesses at least one nontrivial homoclinic solution.

If \(G(t)\) is even in t, we can obtain the following multiplicity result.

Theorem 1.2

Suppose (2), \((L'')\), \((L_{1})\), \((L_{2})\), \((W_{1})\)–\((W_{5})\) and \(G(-x)=G(x)\) for all \(x\in \mathbb{R}^{N}\). Then problem (1) possesses infinitely many solutions.

By a condition similar to \((H_{2})\), we obtain the following.

Theorem 1.3

Suppose (2), \((L'')\), \((L_{1})\), \((L_{2})\), \((W_{2})\)–\((W_{4})\), \((\mathit {SQ})\) and the following condition holds:

\((W_{6})\) :

\(\frac{\widetilde{G}(x)}{G(x)} \vert x \vert ^{2}\rightarrow +\infty \) as \(\vert x \vert \rightarrow \infty \).

Then problem (1) possesses at least one nontrivial homoclinic solution.

Theorem 1.4

Suppose (2), \((L'')\), \((L_{1})\), \((L_{2})\), \((W_{2})\)–\((W_{4})\), \((W_{6})\), \((\mathit {SQ})\) and \(G(-x)=G(x)\) for all \(x\in \mathbb{R}^{N}\). Then problem (1) possesses infinitely many solutions.

Remark 1

It can be seen that \(a(t)\) and \(l(t)\) are positive in our theorems and the results in [28]. Then we can deduce that \((L_{1})\) is weaker than \((L''')\). The reader can see [2] for more details.

Remark 2

Condition \((H_{5})\) does not hold in our theorems since \(a(t)\) can go to infinity as \(\vert t \vert \rightarrow \infty \).

Remark 3

Condition \((W_{1})\) implies \((\mathit {SQ})\).

2 Preliminaries

Set

$$\begin{aligned} E:= \biggl\{ u\in H^{1}\bigl(\mathbb{R},\mathbb{R}^{N} \bigr): \int_{\mathbb{R}}\bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{2}+\bigl(L(t)u(t),u(t)\bigr)\bigr)\,dt< \infty \biggr\} \end{aligned}$$

with the inner product

$$\begin{aligned} (u,v)_{E}= \int_{\mathbb{R}}\bigl(\bigl(\dot{u}(t),\dot{v}(t)\bigr)+ \bigl(L(t)u(t),u(t)\bigr)\bigr)\,dt. \end{aligned}$$

Let \(L_{a}^{2}(\mathbb{R},\mathbb{R}^{N})\) be the weighted space of measurable functions \(u:\mathbb{R}\rightarrow \mathbb{R}^{N}\) under the norm

$$\begin{aligned} \Vert u \Vert _{L_{a}^{2}}= \biggl( \int_{\mathbb{R}}a(t) \bigl\vert u(t) \bigr\vert ^{2}\,dt \biggr) ^{1/2}. \end{aligned}$$

Throughout this paper, \(C>0\) denotes a universal positive constant. Note that under condition \((L_{1})\)

$$\begin{aligned} E\subset H^{1}\bigl(\mathbb{R},\mathbb{R}^{N}\bigr)\subset L^{p}\bigl(\mathbb{R}, \mathbb{R}^{N}\bigr) \end{aligned}$$

for any \(p\in [2,+\infty ]\) with the embedding being continuous, which implies that there is a positive constant \(C_{\infty }>0\) such that the following inequality holds:

$$\begin{aligned} \max \bigl\{ \Vert u \Vert _{2}, \Vert u \Vert _{\infty } \bigr\} \leq C_{\infty } \Vert u \Vert . \end{aligned}$$
(3)

First, we show a compact embedding theorem.

Lemma 2.1

Under conditions \((L_{1})\)–\((L_{2})\) and \((L'')\), the embedding \(E\hookrightarrow L_{a}^{2}(\mathbb{R},\mathbb{R}^{N})\) is continuous and compact.

Proof

Suppose that \(\{u_{n}\}\) is a bounded sequence in E. Then there exists \(M_{0}>0\) such that \(\Vert u_{n} \Vert \leq M_{0}\). Hence there exists a weak convergent subsequence, still denoted by \(\{u_{n}\}\), such that \(u_{n}\rightharpoonup u_{0}\) in E. Assuming \(v_{n}=u_{n}-u_{0}\), we obtain that \(\{v_{n}\}\) is a bounded sequence in E and \(v_{n}\rightharpoonup 0\) in E. Next, we show that \(v_{n}\rightarrow 0\) in \(L_{a}^{2}(\mathbb{R},\mathbb{R}^{N})\). It follows from the Sobolev compact embedding theorem that \(v_{n}\rightarrow 0\) in \(L^{1}(B_{R}(0),\mathbb{R}^{N})\) for any \(R>0\). Choose \(\{s_{i}\} \subset \mathbb{R}\) such that \(\mathbb{R}\subset \bigcup_{i=1}^{ \infty }B_{r_{0}}(s_{i})\) and each \(t\in \mathbb{R}\) is contained by two such intervals at most. Set

$$\begin{aligned} A(M,R)= \biggl\{ t\in B_{R}^{c}(0):\frac{l(t)}{a(t)}\ngeq M \biggr\} \end{aligned}$$

and

$$\begin{aligned} B(M,R)= \biggl\{ t\in B_{R}^{c}(0):\frac{l(t)}{a(t)}\geq M \biggr\} . \end{aligned}$$

On the one hand, we have

$$\begin{aligned} \int_{B(M,R)}a(t) \bigl\vert v_{n}(t) \bigr\vert ^{2}\,dt\leq \frac{1}{M} \int_{B(M,R)}l(t) \bigl\vert v _{n}(t) \bigr\vert ^{2}\,dt\leq \frac{ \Vert v_{n} \Vert ^{2}}{M}\leq \frac{2M_{0}}{M}. \end{aligned}$$
(4)

On the other hand, let \(\varepsilon_{R}=\sup_{s_{i}} ( \operatorname{meas} ( A(M,R) \bigcap B_{r_{0}}(s_{i}) ) ) \), we obtain

$$\begin{aligned} \int_{A(M,R)}a(t) \bigl\vert v_{n}(t) \bigr\vert ^{2}\,dt \leq & \sum_{i=1}^{\infty } \int_{A(M,R)\bigcap B_{r_{0}}(s_{i})}a(t) \bigl\vert v_{n}(t) \bigr\vert ^{2}\,dt \\ \leq & \varepsilon_{R}^{\frac{1}{2}}\sum _{i=1}^{\infty } \biggl( \int_{A(M,R) \bigcap B_{r_{0}}(s_{i})}a^{2}(t) \bigl\vert v_{n}(t) \bigr\vert ^{4}\,dt \biggr) ^{\frac{1}{2}} \\ \leq & A\varepsilon_{R}^{\frac{1}{2}}\sum _{i=1}^{\infty } \biggl( \int_{A(M,R) \bigcap B_{r_{0}}(s_{i})}l^{2}(t) \bigl\vert v_{n}(t) \bigr\vert ^{4}\,dt \biggr) ^{\frac{1}{2}} \\ \leq & AC\varepsilon_{R}^{\frac{1}{2}}\sum _{i=1}^{\infty } \int_{B_{r_{0}}(s_{i})}\bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{2}+\bigl(L(t)u(t),u(t)\bigr)\bigr)\,dt \\ \leq & 2AC\varepsilon_{R}^{\frac{1}{2}} \Vert u \Vert ^{2} \end{aligned}$$
(5)

for some \(C>0\). It follows from \((L_{2})\) that \(\int_{A(M,R)}a(t) \vert v _{n}(t) \vert ^{2}\,dt\rightarrow 0\) as \(R\rightarrow \infty \), which implies that

$$\begin{aligned} \int_{B_{R}^{c}(0)}a(t) \bigl\vert v_{n}(t) \bigr\vert ^{2}\,dt =& \int_{A(M,R)}a(t) \bigl\vert v_{n}(t) \bigr\vert ^{2}\,dt+ \int_{B(M,R)}a(t) \bigl\vert v_{n}(t) \bigr\vert ^{2}\,dt \\ \leq &\frac{2M_{0}}{M}+2d_{0}C\varepsilon_{R}^{\frac{1}{2}} \Vert u \Vert ^{2} \\ \rightarrow &0 \quad \text{as } \min \{M,R\}\rightarrow \infty . \end{aligned}$$

Then we can deduce that \(v_{n}\rightarrow 0\) in \(L_{a}^{2}(\mathbb{R}, \mathbb{R}^{N})\). Hence the embedding \(E\hookrightarrow L_{a}^{2}( \mathbb{R},\mathbb{R}^{N})\) is continuous and compact. □

From Lemma 2.1, we can easily deduce that there is a positive constant \(K>0\) such that the following inequality holds:

$$\begin{aligned} \Vert u \Vert _{L_{a}^{2}}\leq K \Vert u \Vert . \end{aligned}$$
(6)

Lemma 2.2

Suppose that conditions \((L'')\), (2), \((L_{1})\), and \((W_{4})\) hold, then we have \(\nabla G(u_{k})\rightarrow \nabla G(u)\) in \(L_{a}^{2}(\mathbb{R},\mathbb{R}^{N})\) if \(u_{k}\rightharpoonup u\) in E.

Proof

The proof of this lemma is similar to that of Lemma 2.3 in [28]. □

The corresponding functional of (1) is defined by

$$\begin{aligned} I(u)=\frac{1}{2} \Vert u \Vert ^{2}- \int_{\mathbb{R}}W\bigl(t,u(t)\bigr)\,dt . \end{aligned}$$
(7)

Lemma 2.3

Suppose \((L'')\), (2), \((L_{1})\), and \((W_{4})\) hold, then \(I\in C^{1}(E,\mathbb{R})\) is w.l.s.c. and

$$\begin{aligned} \bigl\langle I^{\prime }(u),v \bigr\rangle = \int_{\mathbb{R}}\bigl( \bigl(\dot{u}(t), \dot{v}(t)\bigr)+ \bigl(L(t)u(t),v(t)\bigr)\bigr)\,dt- \int_{\mathbb{R}}\bigl(\nabla W\bigl(t,u(t)\bigr),v(t)\bigr))\,dt \quad \forall u, v\in E. \end{aligned}$$

Proof

The proof is similar to Lemma 2.4 in [28]. □

From Lemma 2.3, we obtain

$$\begin{aligned} \bigl\langle I^{\prime }(u),u \bigr\rangle = \Vert u \Vert ^{2}- \int_{\mathbb{R}}\bigl( \nabla W\bigl(t,u(t)\bigr),u(t)\bigr)\,dt. \end{aligned}$$
(8)

The following theorem is useful in proving the multiplicity of homoclinic solutions for problem (1).

Lemma 2.4

(see [19], Theorem 9.12)

Let \(\mathcal{B}\) be an infinite dimensional real Banach space, and let \(I\in C^{1}(\mathcal{B}, \mathbb{R})\) be even, satisfy the \((\mathit {PS})\) condition and \(I(0)=0\). If \(\mathcal{B}=V\oplus X\), where V is finite dimensional and I satisfies the following:

\((A_{1})\) :

There are constants ϱ, \(\alpha >0\) such that \(I|_{\partial B_{\varrho \bigcap X}}\geq \alpha \); and

\((A_{2})\) :

For each finite dimensional subspace \(\widetilde{E}\subset \mathcal{B}\), there is \(r=r(\widetilde{E})\) such that \(I\leq 0\) on \(\widetilde{E}\setminus B_{r(\widetilde{E})}\).

Then I has an unbounded sequence of critical values.

3 Proof of Theorem 1.1

The proof of Theorem 1.1 is divided into several lemmas.

Lemma 3.1

Suppose that \((\mathit {SQ})\) and \((W_{3})\) hold, then \(G(x)\geq 0\) for all \(x\in \mathbb{R}\).

Proof

The proof of this lemma is similar to that of Lemma 7 in [15]. □

Definition 3.1

A sequence \(\{u_{n}\}\subset X\), where X is a real Banach space, is called a Cerami ((C) for short) sequence of functional I if \(\{I(u_{n})\}\) is bounded and \(\Vert I'(u_{n}) \Vert (1+ \Vert u_{n} \Vert )\rightarrow 0\) as \(n\rightarrow \infty \).

Definition 3.2

A functional I is said to satisfy the (C) condition if any (C) sequence of I possesses a convergent subsequence.

Lemma 3.2

Suppose that the conditions of Theorem 1.1 hold, then I satisfies the \((C)\) condition.

Proof

Assume that \(\{u_{n}\} \subset E\) is a sequence such that \(\{I(u_{n})\}\) is bounded and \(\Vert I'(u_{n}) \Vert (1+ \Vert u_{n} \Vert ) \rightarrow 0\) as \(n \rightarrow \infty \). Then, for some \(M_{1}> 0\), it follows

$$\begin{aligned} \bigl\vert I(u_{n}) \bigr\vert \leq M_{1}, \quad\quad \bigl\Vert I'(u_{n}) \bigr\Vert \bigl(1+ \Vert u_{n} \Vert \bigr) \leq M_{1}. \end{aligned}$$
(9)

Next we show that \(\{u_{n}\}\) is bounded in E. Assuming \(\Vert u_{n} \Vert \rightarrow +\infty \) as \(n\rightarrow \infty \), set \(w_{n}=\frac{u _{n}}{ \Vert u_{n} \Vert }\), then \(\Vert w_{n} \Vert =1\), which implies that there exists a subsequence of \(\{w_{n}\}\), still denoted by \(\{w_{n}\}\), such that \(w_{n}\rightharpoonup w_{0}\) in E. By (7) and (9), we get

$$\begin{aligned} \biggl\vert \int_{\mathbb{R}}\frac{W(t,u_{n}(t))}{ \Vert u_{n} \Vert ^{2}}\,dt- \frac{1}{2} \biggr\vert = \biggl\vert -\frac{I(u_{n})}{ \Vert u_{n} \Vert ^{2}} \biggr\vert \leq \frac{M_{1}}{ \Vert u_{n} \Vert ^{2}}, \end{aligned}$$
(10)

which implies that

$$\begin{aligned} \biggl\vert \int_{\mathbb{R}}\frac{W(t,u_{n}(t))}{ \Vert u_{n} \Vert ^{2}}\,dt \biggr\vert \leq 1 \end{aligned}$$
(11)

for n large enough. The following discussion is divided into two cases.

Case 1: \(w_{0}\not \equiv 0\). Let \(\Omega =\{t\in \mathbb{R}\mid \vert w _{0}(t) \vert >0 \}\). We can see that \(\operatorname{meas}(\Omega )>0\). Then there exists \(\chi >0\) such that \(\operatorname{meas}(\Lambda )>0\), where \(\Lambda =\Omega \bigcap B_{\chi }(0)\). Since \(\Vert u_{n} \Vert \rightarrow +\infty \) as \(n\rightarrow \infty \) and \(\vert u_{n}(t) \vert = \vert w_{n}(t) \vert \cdot \Vert u_{n} \Vert \), we have \(\vert u_{n}(t) \vert \rightarrow +\infty \) as \(n\rightarrow \infty \) for a.e. \(t\in \Lambda \). Let \(a_{1}=\inf_{t\in B_{\chi }(0)}a(t)>0\). By (2), \((W_{2})\), \((W_{4})\), Lemma 3.1, and Fatou’s lemma, we can obtain

$$\begin{aligned} \lim_{n\rightarrow \infty } \int_{\mathbb{R}}\frac{a(t)G(u_{n}(t))}{ \Vert u _{n} \Vert ^{2}}\,dt \geq & a_{1}\lim _{n\rightarrow \infty } \int_{\Lambda }\frac{G(u_{n}(t))}{ \vert u_{n}(t) \vert ^{2}} \bigl\vert w_{n}(t) \bigr\vert ^{2}\,dt { } \\ \geq & a_{1}\lim_{n\rightarrow \infty } \int_{\Lambda }\frac{G(u _{n}(t))}{ \vert u_{n}(t) \vert ^{2}} \bigl\vert w_{n}(t) \bigr\vert ^{2}\,dt{ } \\ \geq & a_{1}\lim_{n\rightarrow \infty } \int_{\Lambda }\frac{G(u _{n}(t))}{ \vert u_{n}(t) \vert ^{2}} \bigl\vert w_{n}(t) \bigr\vert ^{2}\,dt{ } \\ =& +\infty , \end{aligned}$$

which contradicts (11).

Case 2: \(w_{0}\equiv 0\). Set a sequence \(\{T_{n}\}\subset [0,1]\) such that \(I(T_{n}u_{n})=\max_{T\in [0,1]}I(Tu_{n})\). By Lemma 3.1, \((W_{2})\), \((W_{5})\), and Lemma 2.1, we obtain

$$\begin{aligned} 0 \leq & \int_{\mathbb{R}}a(t)G\bigl(4\sqrt{\theta M_{1}}w_{n}(t) \bigr)\,dt \\ \leq & d_{1} \biggl( 16\theta M_{1} \int_{\mathbb{R}}a(t) \bigl\vert w_{n}(t) \bigr\vert ^{2}\,dt+(4\sqrt{ \theta M_{1}})^{\zeta } \int_{\mathbb{R}}a(t) \bigl\vert w_{n}(t) \bigr\vert ^{\zeta }\,dt \biggr) \\ =&d_{1} \bigl( 16\theta M_{1}+(4\sqrt{\theta M_{1}})^{\zeta } \Vert w_{n} \Vert _{\infty }^{\zeta -2} \bigr) \int_{\mathbb{R}}a(t) \bigl\vert w_{n}(t) \bigr\vert ^{2}\,dt \\ =&d_{1} \bigl( 16\theta M_{1}+(4\sqrt{\theta M_{1}})^{\zeta }C_{ \infty }^{\zeta -2} \bigr) \int_{\mathbb{R}}a(t) \bigl\vert w_{n}(t) \bigr\vert ^{2}\,dt\rightarrow 0 \quad \text{as } n\rightarrow \infty , \end{aligned}$$

which implies that

$$\begin{aligned} \int_{\mathbb{R}}a(t)G\bigl(4\sqrt{\theta M_{1}}w_{n}(t) \bigr)\,dt\rightarrow 0 \quad \text{as } n\rightarrow \infty . \end{aligned}$$
(12)

By the definition of \(T_{n}\) and (12), for n large enough, we have

$$\begin{aligned} I(T_{n}u_{n}) \geq & I \biggl( \frac{4\sqrt{\theta M_{1}}}{ \Vert u_{n} \Vert }u _{n} \biggr) \\ =&I(4\sqrt{\theta M_{1}}w_{n}) \\ =&\frac{1}{2} \Vert 4\sqrt{\theta M_{1}}w_{n} \Vert ^{2}- \int_{\mathbb{R}}a(t)G\bigl(4\sqrt{ \theta M_{1}}w_{n}(t) \bigr)\,dt \\ =&8\theta M_{1}- \int_{\mathbb{R}}a(t)G\bigl(4\sqrt{\theta M_{1}}w_{n}(t) \bigr)\,dt \\ \geq &4\theta M_{1}. \end{aligned}$$

Then we obtain

$$\begin{aligned} & \Vert T_{n}u_{n} \Vert ^{2}- \int_{\mathbb{R}}a(t) \bigl(\nabla G\bigl(T_{n}u_{n}(t) \bigr),T _{n}u_{n}(t)\bigr)\,dt \\ &\quad =\bigl\langle I^{\prime }\bigl(T_{n}u_{n}(t) \bigr),T_{n}u_{n}(t)\bigr\rangle =T_{n} \frac{dI(Tu_{n}(t))}{dT} \bigg| _{T=T_{n}}=0. \end{aligned}$$
(13)

Hence, it follows from (13) and \((W_{1})\) that

$$\begin{aligned} & \int_{\mathbb{R}}a(t) \biggl( \frac{1}{2}\bigl(\nabla G \bigl(u_{n}(t)\bigr),u_{n}(t)\bigr)-G\bigl(u _{n}(t)\bigr) \biggr) \,dt \\ &\quad =\frac{1}{2} \int_{\mathbb{R}}a(t)\widetilde{G}\bigl(u_{n}(t)\bigr)\,dt \\ &\quad \geq \frac{1}{2\theta } \int_{\mathbb{R}}a(t)\widetilde{G}\bigl(T_{n}u _{n}(t)\bigr)\,dt \\ &\quad =\frac{1}{\theta } \int_{\mathbb{R}} \biggl( \frac{1}{2}a(t) \bigl(\nabla G\bigl(T _{n}u_{n}(t)\bigr),T_{n}u_{n}(t) \bigr)-a(t)G\bigl(T_{n}u_{n}(t)\bigr) \biggr) \,dt \\ &\quad =\frac{1}{\theta } \biggl( \frac{1}{2} \Vert T_{n}u_{n} \Vert ^{2}- \int_{ \mathbb{R}}a(t)G\bigl(T_{n}u_{n}(t)\bigr) \,dt \biggr) \\ &\quad =\frac{1}{\theta }I(T_{n}u_{n}), \end{aligned}$$

which implies that

$$\begin{aligned} \int_{\mathbb{R}}a(t) \biggl( \frac{1}{2}\bigl(\nabla G \bigl(u_{n}(t)\bigr),u_{n}(t)\bigr)-G\bigl(u _{n}(t)\bigr) \biggr) \,dt\geq 4 M_{1} \quad \text{for $n$ large enough}. \end{aligned}$$
(14)

However, we can deduce from (9) that

$$\begin{aligned} \biggl\vert \int_{\mathbb{R}}a(t) \biggl( \frac{1}{2}\bigl(\nabla G \bigl(u_{n}(t)\bigr),u_{n}(t)\bigr)-G\bigl(u _{n}(t)\bigr) \biggr) \,dt \biggr\vert = \bigl\vert 2I(u_{n})-\bigl\langle I'(u_{n}),u_{n} \bigr\rangle \bigr\vert \leq 3 M_{1} \end{aligned}$$

for all \(n\in \mathbb{N}\), which contradicts (14). Hence \(\{u_{n}\}\) is bounded in E. Going if necessary to a subsequence, we can assume that \(u_{n} \rightharpoonup u\) in E, which yields

$$\begin{aligned} \bigl\langle I'(u_{n})-I'(u),u_{n}-u \bigr\rangle \rightarrow 0 \quad \text{as } n\rightarrow \infty \end{aligned}$$

and it follows from Lemma 2.1, Hölder’s inequality, (6), and Lemma 2.2 that

$$\begin{aligned} \biggl\vert \int_{\mathbb{R}}a(t) \bigl(\nabla G\bigl(u_{n}(t)\bigr)- \nabla G\bigl(u(t)\bigr),u_{n}(t)-u(t)\bigr)\,dt \biggr\vert &\leq \bigl\Vert \nabla G(u_{n})-\nabla G(u) \bigr\Vert _{L_{a}^{2}} \Vert u_{n}-u \Vert _{L_{a} ^{2}} \\ &\rightarrow 0 \quad \text{as } n\rightarrow \infty . \end{aligned}$$

Hence, we conclude that

$$\begin{aligned} & \Vert u_{n}-u \Vert ^{2} \\ &\quad =\bigl\langle I'(u_{n})-I'(u),u_{n}-u \bigr\rangle + \int_{\mathbb{R}}a(t) \bigl( \nabla G\bigl(u_{n}(t)\bigr)- \nabla G\bigl(u(t)\bigr),u_{n}(t)-u(t)\bigr)\,dt \\ &\quad \rightarrow 0 \quad \text{as } n\rightarrow \infty . \end{aligned}$$

We finish the proof. □

Subsequently, we need the following lemmas to prove Theorem 1.1.

Lemma 3.3

Suppose that (2), \((L'')\), \((L_{1})\), \((L_{2})\), \((W_{1})\), and \((W_{3})\) hold, then there exist constants ϱ, \(\alpha > 0\) such that \(I|_{\partial B_{\varrho }(0)} \geq \alpha \).

Proof

By \((W_{3})\), for any \(\varepsilon >0\), there is \(\sigma >0\) such that

$$\begin{aligned} \bigl\vert G(x) \bigr\vert \leq \varepsilon \vert x \vert ^{2} \end{aligned}$$
(15)

for all \(\vert x \vert \leq \sigma \). For \(\varepsilon_{1}=\frac{1}{4A}\), there exists \(\delta_{1}>0\) such that (15) holds. Set

$$\begin{aligned} \varrho =C_{\infty }^{-1}\delta_{1},\quad\quad \alpha = \frac{1}{4}\varrho^{2}. \end{aligned}$$

By (7), (15), (2), \((L_{2})\), and \((W_{1})\), for any \(\Vert u \Vert =\varrho \), we obtain

$$\begin{aligned} I(u) =& \frac{1}{2} \Vert u \Vert ^{2}- \int_{\mathbb{R}}W\bigl(t,u(t)\bigr)\,dt{ } \\ =& \frac{1}{2} \Vert u \Vert ^{2}- \int_{\mathbb{R}}a(t)G\bigl(u(t)\bigr)\,dt{ } \\ \geq & \frac{1}{2} \Vert u \Vert ^{2}-\frac{1}{4A} \int_{\mathbb{R}}a(t) \bigl\vert u(t) \bigr\vert ^{2}\,dt { } \\ \geq & \frac{1}{2} \Vert u \Vert ^{2}-\frac{1}{4} \int_{\mathbb{R}}l(t) \bigl\vert u(t) \bigr\vert ^{2}\,dt { } \\ \geq & \frac{1}{4} \Vert u \Vert ^{2}. \end{aligned}$$
(16)

By the definition of ϱ and α, (16) implies \(I|_{\partial B_{\varrho }(0)} \geq \alpha \). □

Lemma 3.4

Suppose that (2), \((L'')\), \((W_{2})\), and \((W_{4})\) hold, then there exists \(\bar{e}\in E\) such that \(\Vert \bar{e} \Vert >\varrho \) and \(I(\bar{e})\leq 0\), where ϱ is defined in Lemma 3.3.

Proof

Set \(e_{0} \in C^{\infty }_{0}(-1,1)\) with \(\Vert e_{0} \Vert =1\). Let \(a_{2}=\min_{t\in B_{1}(0)}a(t)\) and \(a_{3}=\max_{t\in B_{1}(0)}a(t)\). For \(\beta >\frac{1}{2a_{2}\int^{1} _{-1} \vert e_{0}(t) \vert ^{2}\,dt}\), it follows from \((W_{4})\) that there exists \(\xi >0\) such that

$$\begin{aligned} G(x)\geq \beta \vert x \vert ^{2} \end{aligned}$$

for all \(\vert x \vert > \xi \). By Lemma 3.1, we have

$$\begin{aligned} G(x)\geq \beta \bigl( \vert x \vert ^{2}-\xi^{2}\bigr) \end{aligned}$$
(17)

for all \(x\in \mathbb{R}^{N}\). By (2), (7), and (17), for every \(\eta \in \mathbb{R}\setminus \{0\}\), we have

$$\begin{aligned} I(\eta e_{0}) =&\frac{\eta^{2}}{2} \Vert e_{0} \Vert ^{2}- \int^{1}_{-1}W\bigl(t, \eta e_{0}(t)\bigr) \,dt{ } \\ =&\frac{\eta^{2}}{2} \Vert e_{0} \Vert ^{2}- \int^{1}_{-1}a(t)G\bigl(\eta e_{0}(t)\bigr) \,dt { } \\ \leq & \frac{\eta^{2}}{2}-a_{2}\beta \eta^{2} \int^{1}_{-1} \bigl\vert e _{0}(t) \bigr\vert ^{2}\,dt+2a_{3}\beta \xi^{2}{ } \\ =& \biggl( \frac{1}{2}-a_{2}\beta \int^{1}_{-1} \bigl\vert e_{0}(t) \bigr\vert ^{2}\,dt \biggr) \eta^{2}+2a_{3}\beta \xi^{2}, \end{aligned}$$

which implies that

$$\begin{aligned} I(\eta e_{0})\rightarrow -\infty \quad \text{as } \eta \rightarrow + \infty . \end{aligned}$$

Then there exists \(\eta_{0} \in \mathbb{R} \setminus \{0\} \) such that \(\Vert \eta_{0} e_{0} \Vert >\varrho \) and \(I(\eta_{0} e_{0})<0\). Letting \(\bar{e}(t)=\eta_{0} e_{0}(t)\), we finish the proof. □

Proof of Theorem 1.1

It is known that a deformation lemma can be proved when the usual \((\mathit {PS})\) condition is replaced with condition \((C)\). From the above proofs and the mountain pass theorem (see [19], Theorem 2.2), I possesses a critical value \(c \geq \alpha \) and a critical point \(u_{0}\) such that \(I(u_{0})=c\), which means problem (1) has at least one nontrivial homoclinic solution. □

4 Proof of Theorem 1.3

Lemma 4.1

Suppose that the conditions of Theorem 1.2 hold, then I satisfies the \((C)\) condition.

Proof

Assuming that \(\{u_{n}\} \subset E\) is a \((C)\) sequence of I, there exists a constant \(M_{2} > 0\) such that

$$\begin{aligned} \bigl\vert I(u_{n}) \bigr\vert \leq M_{2}, \quad\quad \bigl\Vert I'(u_{n}) \bigr\Vert \bigl(1+ \Vert u_{n} \Vert \bigr) \leq M_{2}. \end{aligned}$$
(18)

Now we show the boundedness of \(\{u_{n}\}\). Arguing in an indirect way, we assume that \(\Vert u_{n} \Vert \rightarrow +\infty \) as \(n\rightarrow \infty \). Set \(w_{n}=\frac{u_{n}}{ \Vert u_{n} \Vert }\), then \(\Vert w_{n} \Vert =1\), which implies that there exists a subsequence of \(\{w_{n}\}\), still denoted by \(\{w_{n}\}\), such that \(w_{n}\rightharpoonup w_{0}\) in E. Similar to the proof of Lemma 3.2, we have

$$\begin{aligned} \int_{\mathbb{R}}\frac{W(t,u_{n})}{ \Vert u_{n} \Vert ^{2}}\,dt\rightarrow \frac{1}{2} \quad \text{as } n\rightarrow \infty . \end{aligned}$$
(19)

Case 1: \(w_{0}\not \equiv 0\). The proof is similar to the proof of Case 1 in Lemma 3.2.

Case 2: \(w_{0}\equiv 0\). Let \(\varepsilon =1\), then there exists \(\sigma_{0}>0\) such that (15) holds for all \(\vert x \vert \leq \sigma_{0}\). By \((W_{6})\), we obtain that for any \(B>0\), there exists \(r_{\infty }>0\) such that, for all \(\vert x \vert \geq r_{\infty }\), we have

$$\begin{aligned} \frac{\widetilde{G}(x)}{G(x)} \vert x \vert ^{2}\geq B. \end{aligned}$$
(20)

It follows from (7), (8), (20), \((W_{1})\), and \((\mathit {SQ})\) that

$$\begin{aligned} 0 \leq & \int_{\mathbb{R}}\frac{W(t,u_{n})}{ \Vert u_{n} \Vert ^{2}}\,dt{ } \\ \leq & \int_{\{t\in \mathbb{R}\mid \vert u_{n} \vert > r_{\infty }\}}\frac{W(t,u _{n})}{ \vert u_{n}(t) \vert ^{2}} \bigl\vert w_{n}(t) \bigr\vert ^{2}\,dt+ \int_{\{t\in \mathbb{R}\mid \vert u_{n} \vert \leq \sigma_{0}\}}\frac{W(t,u_{n})}{ \Vert u _{n} \Vert ^{2}}\,dt{ } \\ & {} + \int_{\{t\in \mathbb{R}\mid \sigma_{0}\leq \vert u_{n} \vert \leq r_{\infty }\}}\frac{W(t,u _{n})}{ \Vert u_{n} \Vert ^{2}}\,dt{ } \\ \leq & \Vert w_{n} \Vert _{L^{\infty }}^{2} \int_{\{t\in \mathbb{R}\mid \vert u_{n} \vert > r_{\infty }\}}\frac{a(t)G(u_{n})}{ \vert u _{n}(t) \vert ^{2}}\,dt+ \int_{\{t\in \mathbb{R}\mid \vert u_{n} \vert \leq \sigma_{0}\}}a(t) \bigl\vert w _{n}(t) \bigr\vert ^{2}\,dt{ } \\ & {} + \int_{\{t\in \mathbb{R}\mid \sigma_{0}\leq \vert u_{n} \vert \leq r_{\infty }\}}\frac{a(t)G(u _{n}) \vert u_{n}(t) \vert ^{2}}{\sigma_{0}^{2} \Vert u_{n} \Vert ^{2}}\,dt{ } \\ \leq & \frac{ \Vert w_{n} \Vert _{L^{\infty }}^{2}}{B} \int_{\{t\in \mathbb{R}\mid \vert u_{n} \vert > r_{\infty }\}}a(t) \bigl(\bigl(\nabla G(u_{n}),u _{n}\bigr) -2G(u_{n})\bigr)\,dt+ \Vert w_{n} \Vert _{L_{a}^{2}}^{2}{ } \\ & {} +\frac{\max_{\sigma_{0}\leq \vert x \vert \leq \rho_{\infty }} \vert G(x) \vert }{ \sigma^{2}_{0}} \int_{\{t\in \mathbb{R}\mid \sigma_{0}\leq \vert u_{n} \vert \leq r_{\infty }\}}a(t) \bigl\vert w _{n}(t) \bigr\vert ^{2}\,dt{ } \\ \leq & \frac{ \Vert w_{n} \Vert _{L^{\infty }}^{2}}{B} \int_{\{t\in \mathbb{R}\mid \vert u_{n} \vert > r_{\infty }\}}a(t) \bigl(\bigl(\nabla G(u_{n}),u _{n}\bigr) -2G(u_{n})\bigr)\,dt{ } \\ & {} + \biggl( 1+\frac{\max_{\sigma_{0}\leq \vert x \vert \leq \rho_{\infty }} \vert G(x) \vert }{ \sigma^{2}_{0}} \biggr) \Vert w_{n} \Vert _{L_{a}^{2}}^{2}{ } \\ \leq & \frac{ \Vert w_{n} \Vert _{L^{\infty }}^{2}}{B} \bigl( 2I(u_{n})- \bigl\langle I'(u_{n}),u_{n}\bigr\rangle \bigr) + \biggl( 1+\frac{ \max_{\sigma_{0}\leq \vert x \vert \leq \rho_{\infty }} \vert G(x) \vert }{\sigma^{2}_{0}} \biggr) \Vert w_{n} \Vert _{L_{a}^{2}}^{2}{ } \\ \leq & \frac{3M_{2}C_{\infty }^{2}}{B}+ \biggl( 1+\frac{ \max_{\sigma_{0}\leq \vert x \vert \leq \rho_{\infty }} \vert G(x) \vert }{\sigma^{2}_{0}} \biggr) \Vert w_{n} \Vert _{L_{a}^{2}}^{2}. \end{aligned}$$

By the arbitrariness of B and Lemma 3.1, we have

$$\begin{aligned} \int_{\mathbb{R}}\frac{W(t,u_{n})}{ \Vert u_{n} \Vert ^{2}}\,dt< \frac{1}{4} \end{aligned}$$
(21)

for n large enough, which contradicts (19). Hence \(\Vert u_{n} \Vert \) is bounded in E. Similar to the proof of Lemma 3.2, we see that I satisfies the \((C)\) condition. □

The following proof is similar to that of Theorem 1.1.

5 Proof of Theorems 1.2 and 1.4

As we know, Lemma 2.4 still holds when we replace the \((\mathit {PS})\) condition with the \((C)\) condition. From Lemmas 3.2 and 4.1, we can see that I satisfies the \((C)\) condition under the conditions of Theorems 1.2 and 1.4, respectively. Next, we only show that I satisfies the geometric conditions of Lemma 2.4. Obviously, Lemma 3.3 holds under the conditions of Theorems 1.2 and 1.4, respectively. If we take \(V=\{0\}\) and \(X=E\), we can see there are constants \(\varrho_{1}\), \(\alpha_{1}>0\) such that \(I|_{\partial B_{\varrho_{1}\bigcap X}} \geq \alpha_{1}\) by Lemma 3.2, which means \((A_{1})\) is fulfilled. Subsequently, we show that I satisfies \((A_{2})\).

Lemma 5.1

Suppose that (2), \((L'')\), \((L_{1})\), \((L_{2})\), \((W_{3})\), and \((W_{4})\) hold, then I satisfies \((A_{2})\).

Proof

Let \(\widetilde{E} \subset E\) be a finite dimensional subspace. For any \(u\in \widetilde{E}\setminus \{0\}\) and \(\vartheta >0\), set

$$\begin{aligned} \Gamma_{\vartheta }(u)=\bigl\{ t\in \mathbb{R}: \bigl\vert u(t) \bigr\vert \geq \vartheta \Vert u \Vert \bigr\} . \end{aligned}$$

Similar to Lemma 6.2 in [29], there exists \(\vartheta_{0}>0\) such that

$$\begin{aligned} \operatorname{meas} \bigl( \Gamma_{\vartheta_{0}}(u) \bigr) \geq \vartheta_{0} \end{aligned}$$
(22)

for all \(u\in \widetilde{E}\setminus \{0\}\). It is easy to see that there exists \(\rho >0\) such that \(\operatorname{meas}\Upsilon_{\vartheta_{0}}(u)> \frac{1}{2}\vartheta_{0}\) for any \(u\in \widetilde{E}\setminus \{0\}\), where \(\Upsilon_{\vartheta_{0}}(u)=\Gamma_{\vartheta_{0}}(u)\bigcap B _{\rho }(0)\). Set \(a_{4}= \inf_{ \vert t \vert \leq \rho }a(t)>0\). By \((W_{4})\), there exists \(\gamma >0\) such that

$$\begin{aligned} G\bigl(u(t)\bigr)\geq \frac{1}{a_{4}\vartheta_{0}^{3}} \bigl\vert u(t) \bigr\vert ^{2}\geq \frac{1}{a _{4}\vartheta_{0}} \Vert u \Vert ^{2} \end{aligned}$$
(23)

for all \(u\in \widetilde{E}\) and \(t\in \Upsilon_{\vartheta_{0}}(u)\) with \(\Vert u \Vert \geq \gamma \). Then, for any \(u\in \widetilde{E}\setminus B_{ \gamma }\), it follows from (7), \((W_{1})\), \((W_{3})\), and (23) that

$$\begin{aligned} I(u) =&\frac{1}{2} \Vert u \Vert ^{2}- \int_{\mathbb{R}}a(t)G\bigl(u(t)\bigr)\,dt{ } \\ =& \frac{1}{2} \Vert u \Vert ^{2}- \int_{\Upsilon_{\vartheta_{0}}(u)}a(t)G\bigl(u(t)\bigr)\,dt- \int_{\mathbb{R}\setminus \Upsilon_{\vartheta_{0}}(u)}a(t)G\bigl(u(t)\bigr)\,dt { } \\ \leq & \frac{1}{2} \Vert u \Vert ^{2}-a_{4} \int_{\Upsilon_{\vartheta_{0}}(u)}G\bigl(u(t)\bigr)\,dt{ } \\ \leq & \frac{1}{2} \Vert u \Vert ^{2}-\frac{1}{\vartheta_{0}}\operatorname{meas} \bigl( \Upsilon _{\vartheta_{0}}(u) \bigr) \Vert u \Vert ^{2}{ } \\ \leq & \frac{1}{2} \Vert u \Vert ^{2}- \Vert u \Vert ^{2}{ } \\ =& -\frac{1}{2} \Vert u \Vert ^{2}. \end{aligned}$$

Then there exists \(r>\gamma \) such that \(I |_{\widetilde{E}\setminus B_{r}}\leq 0\). □

Since I is even in u, I possesses an unbounded sequence of critical values by Lemma 2.4, which proves Theorems 1.2 and 1.4.