1 Introduction and main results

In this paper, we consider the existence of two nontrivial homoclinic solutions for the following second order Hamiltonian systems:

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

for all \(t \in R\), where \(W\in C^{1}( R\times R^{N}, R)\), L: \(R\rightarrow R^{N^{2}}\) is a matrix-valued function and \(f\in C( R, R^{N})\). A solution \(u(t)\) of problem (1) is homoclinic (to 0) if \(u(t)\rightarrow 0\) as \(t\rightarrow \pm \infty \). Moreover, if \(u(t)\neq 0\), \(u(t)\) is called a nontrivial homoclinic solution. Here and subsequently, \(\nabla W(t,x)\) denotes the gradient with respect to the x variable.

The homoclinic solutions have been proved to be important in studying the behavior of dynamic systems. There have been many papers concerning this topic by using the variational methods since the remarkable results by Ambrosetti and Rabinowitz [1]. Because of the lack of compactness, this problem is more difficult than studying the existence of periodic solutions. In order to get the compactness of embedding theorem back, many conditions have been proposed (see [1–41]). Two kinds of important conditions are periodic and coercive conditions. The periodic condition was introduced by Rabinowitz [19] in 1990 to discuss the existence of homoclinic solutions for problem (1) as the limit of a sequence of subharmonics which are obtained by the mountain pass theorem. The following coercive condition is a classical condition introduced by Rabinowitz and his co-author [20].

\((L')\) :

\(L\in C( R, R^{N^{2}})\) is a symmetric and positively definite matrix for all \(t\in R\) and

$$\begin{aligned} \inf_{\vert x\vert =1}\bigl(L(t)x,x\bigr)\rightarrow +\infty \quad \mbox{as } \vert t\vert \rightarrow \infty. \end{aligned}$$

Condition \((L')\) has been studied by many other mathematicians to deal with the nonperiodic systems. After then, there have been some other coercive conditions introduced by other mathematicians.

By using the variational methods to study problem (1), the growth conditions of \(W(t,x)\) are needed. These conditions are mainly classified into three cases: the superquadratic case, the subquadratic case, and the asymptotically quadratic case. In this paper, we mainly consider the superquadratic case. The following growth condition is a classical superquadratic condition known as the \((AR)\) condition.

\((AR)\) :

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

However, the \((AR)\) condition is so strong that many functions cannot be involved. In order to study problem (1) with different potentials, many other superquadratic conditions are proposed. In 2009, Ding and Lee [8] introduced the following generalized superquadratic condition.

\((GS)\) :

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

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

where

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

Some examples are given to show the difference between \((GS)\) and \((AR)\) conditions. The following superquadratic condition is used by Lv and Tang [14] to obtain infinitely many homoclinic solutions for problem (1) when \(W(t,x)\) is even in x.

\((MC)\) :

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

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

for all \((t,x)\in R\times R^{N}\). Recently, Wu et al. [33] introduced the following condition:

\((SQ)\) :

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

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

With \((SQ)\), the authors obtained the existence of homoclinic solutions for a class of periodic Hamiltonian systems.

In 2018, Wu et al. [32] showed the existence of homoclinic solutions for problem (1) without periodic or even conditions. In 2015, Xu et al. [36] showed the existence of two solutions for problem (1) with a nonzero perturbation. In the same year, Zhang and Yuan [40] obtained two homoclinic solutions for a class of perturbed Hamiltonian systems under the \((AR)\) condition. In this paper, we introduce a new superquadratic condition to study problem (1) with small forcing terms. The following theorem is our main result.

Theorem 1.1

Suppose that W and L satisfy the following conditions:

\((L)\) :

For the smallest eigenvalue of \(L(t)\), i.e., \(l(t)\equiv \inf_{\vert x\vert =1} L(t)x\cdot x\), there exists a constant \(\nu <1\) such that \(l(t)\vert t\vert ^{\nu -2}\rightarrow +\infty \) as \(\vert t\vert \rightarrow \infty \).

\((W1)\) :

\(W(t,0)=0\) for all \(t \in R\) and \(\nabla W(t,x)=o(\vert x\vert )\) as \(\vert x\vert \rightarrow 0\) uniformly for \(t\in R\);

\((W2)\) :

\((\nabla W(t,x),x)\geq 2W(t,x)\geq 0\) for all \((t,x) \in R \times R^{N}\);

\((W3)\) :

\(W(t,x) / \vert x\vert ^{2} \rightarrow +\infty \) as \(\vert x\vert \rightarrow \infty \) uniformly in t;

\((W4)\) :

There exist \(\tau >2\) and \(d_{1}>0\) such that

$$\begin{aligned} \bigl\vert \nabla W(t,x)\bigr\vert \leq d_{1}\bigl(1+\vert x \vert ^{\tau -1}\bigr)\quad \textit{for all } t\in R; \end{aligned}$$
\((W5)\) :

There exist constants \(\mu \geq 1\), \(\lambda_{0}\in (0,1)\), \(d_{2}>0\), and \(r_{\infty }>0\) such that

$$\begin{aligned} \biggl( \frac{1-\lambda^{2}}{2}-\lambda \biggr) \bigl(\nabla W(t,x),x\bigr)+W(t, \lambda x)-W(t,x)\geq -d_{2}\lambda^{\mu }\vert x\vert ^{\mu } \end{aligned}$$

for all \(\lambda \in [0,\lambda_{0}]\), \(\vert x\vert \geq r_{\infty }\), and \(t\in R\).

Then there exists \(\delta >0\) such that, for any \(f\not \equiv 0\) satisfying

$$\begin{aligned} \max_{t\in R}\bigl\vert f(t)\bigr\vert \leq \delta, \end{aligned}$$
(2)

system (1) possesses at least two nontrivial homoclinic solutions.

Remark 1

In Theorem 1.1, the perturbation f is not required to be integrable.

Remark 2

Consider the following example:

$$\begin{aligned} G(x)=\vert x\vert ^{s}+(s-2)\vert x\vert ^{s-\epsilon } \sin^{2}\bigl(\vert x\vert ^{\epsilon }/\epsilon \bigr), \end{aligned}$$
(3)

where \(s>2\) and \(\epsilon \in (0,s-2)\). It is easy to check that (3) satisfies the conditions of Theorem 1.1 and but not the \((AR)\) condition. As we know, Theorem 1.1 is the first result to obtain the existence of two homoclinic solutions for problem (1) without the \((AR)\) condition.

2 Proof of Theorem 1.1

Let A be a self-adjoint extension of the operator \(-(d^{2}/dt^{2})+L(t)\) with the domain \(\mathcal{D}(A)\subset L^{2}( R, R^{N})\). Let \(E=\mathcal{D}(\vert A\vert ^{1/2})\) be the domain of \(\vert A\vert ^{1/2}\) and define on E the inner product and the norm as \((u,w)_{0}=(\vert A\vert ^{1/2}u,\vert A\vert ^{1/2}w)_{2}+(u,w)_{2}\) and \(\Vert u\Vert _{0}=(u,u)^{1/2}\), respectively, where \((\cdot,\cdot)_{2}\) denotes the inner product of \(L^{2}\). Then E is a Hilbert space.

It is known that the spectrum \(\sigma (A)\) consists of eigenvalues numbered in \(\lambda_{1}\leq \lambda_{2} \leq \cdots \rightarrow \infty \), and a corresponding system of eigenfunctions \((e_{n})(Ae _{n} =\lambda_{n}e_{n})\) forms an orthogonal basis in \(L^{2}\). Let \(n^{-}=\#\{i\vert \lambda_{i}<0\}\), \(n^{0}=\#\{i\vert \lambda_{i}=0\}\), and \(\bar{n}=n^{-}+n^{0}\). Set \(E^{-}=\operatorname{span}\{e_{1},\ldots,e_{n^{-}}\}\), \(E^{0}=\operatorname{span}\{e_{n^{-}+1},\ldots,e_{\bar{n}}\}\), and \(E^{+}=\overline{\operatorname{span}\{e_{\bar{n}+1},\ldots \}}\). Then \(E=E^{-}\oplus E^{0}\oplus E ^{+}\). The inner product and the norm on E are introduced as

$$\begin{aligned} (u,w)=\bigl(\vert A\vert ^{1/2}u,\vert A\vert ^{1/2}w \bigr)_{2}+\bigl(u^{0},w^{0}\bigr)_{2},\qquad \Vert u\Vert =(u,u)^{1/2}, \end{aligned}$$

where \(u=u^{-}+u^{0}+u^{+}\) and \(w=w^{-}+w^{0}+w^{+}\in E=E^{-}\oplus E^{0}\oplus E^{+}\). Furthermore, let \(I: E \rightarrow R\) be the functional defined by

$$\begin{aligned} I(u) =& \int_{ R} \biggl( \frac{1}{2}\bigl\vert \dot{u}(t)\bigr\vert ^{2}+\frac{1}{2}\bigl(L(t)u(t),u(t)\bigr)-W\bigl(t,u(t) \bigr)+\bigl(f(t),u(t)\bigr) \biggr)\,dt { } \\ =& \frac{1}{2}\Vert u\Vert ^{2}- \int_{ R}W\bigl(t,u(t)\bigr)\,dt+ \int_{ R}\bigl(f(t),u(t)\bigr)\,dt. \end{aligned}$$
(4)

It is known that the critical points of I in E are the homoclinic solutions of (1). One can easily check that \(I\in C^{1}(E, R)\) and

$$\begin{aligned} \bigl\langle I'(u),v\bigr\rangle =& \int_{ R} \bigl( \bigl(\dot{u}(t),\dot{v}(t)\bigr)+ \bigl(L(t)u(t),v(t)\bigr)-\bigl( \nabla W\bigl(t,u(t)\bigr),v(t)\bigr) \bigr)\,dt \\ &{}+ \int_{ R}\bigl(f(t),v(t)\bigr)\,dt. \end{aligned}$$
(5)

By Lemma 2.2 in [6] we can conclude that E is compactly embedded in \(L^{p}\) for any \(p\in [1,+\infty ]\), which implies that there exists a constant \(C_{p}>0\) such that

$$\begin{aligned} \Vert u\Vert _{L^{p}}\leq C_{p}\Vert u\Vert \quad \mbox{for all } u\in E. \end{aligned}$$
(6)

Lemma 2.1

Suppose that the conditions of Theorem 1.1 hold, then there exist constants α, \(\varrho >0\) such that \(I\vert_{S} \geq \alpha \), where \(S=\{u\in E\vert \Vert u\Vert =\varrho \}\).

Proof

By \((W1)\), for any \(\varepsilon >0\), there exists \(\sigma >0\) such that

$$\begin{aligned} \bigl\vert \nabla W(t,x)\bigr\vert \leq \varepsilon \vert x\vert ,\qquad \vert x\vert \leq \sigma, \quad \forall t \in R. \end{aligned}$$

Then it follows from \((W1)\) and \((W2)\) that

$$\begin{aligned} W(t,x) =&\bigl\vert W(t,x)-W(t,0)\bigr\vert \\ =& \biggl\vert \int^{1}_{0}\bigl(\nabla W(t,sx),x\bigr)\,ds\biggr\vert \\ \leq & \int^{1}_{0}\bigl\vert \nabla W(t,sx)\bigr\vert \vert x\vert \,ds \\ \leq & \int^{1}_{0}\varepsilon \vert sx\vert \vert x \vert \,ds \\ \leq & \varepsilon \vert x\vert ^{2} \end{aligned}$$
(7)

for all \(t\in R\) and \(\vert x\vert \leq \sigma \). Let \(\varepsilon_{0}=\frac{1}{4C _{2}}\), then there exists \(\sigma_{0}>0\) such that (7) holds for all \(t\in R\) and \(\vert x\vert \leq \sigma_{0}\). Set

$$\begin{aligned} \varrho =\frac{\sigma_{0}}{C_{\infty }}, \qquad \alpha =\frac{1}{4}\varrho^{2}>0, \end{aligned}$$

which implies \(0 < \Vert u\Vert _{L^{\infty }} \leq \sigma_{0}\) for all \(u \in S\). Then it follows from the definition of I, (7) and (2) that

$$\begin{aligned} I(u) =& \frac{1}{2}\Vert u\Vert ^{2}- \int_{ R}W\bigl(t,u(t)\bigr)\,dt+ \int_{ R}\bigl(f(t),u(t)\bigr)\,dt { } \\ \geq & \frac{1}{2}\Vert u\Vert ^{2}- \frac{1}{4C_{2}} \int_{ R}\bigl\vert u(t)\bigr\vert ^{2}\,dt- \delta \int_{ R}\bigl\vert u(t)\bigr\vert \,dt{ } \\ \geq & \frac{1}{4}\Vert u\Vert ^{2}-\delta C_{1}\Vert u\Vert . \end{aligned}$$

By the definitions of ϱ and α, there exists \(\delta_{0}>0\) such that \(I\mid_{S} \geq \alpha \) for any f satisfying (2). □

Lemma 2.2

Suppose that the conditions of Theorem 1.1 hold, then there is \(e\in E\) such that \(\Vert e\Vert > \varrho \) and \(I(e) \leq 0\), where ϱ is defined in Lemma 2.1.

Proof

It follows from \((W3)\) that there exist \(T>0\), \(\xi >0\), and \(\varepsilon_{1}>0\) such that

$$\begin{aligned} W(t,x)\geq \biggl( \frac{\pi^{2}}{2T^{2}}+\varepsilon_{1} \biggr) \vert x \vert ^{2} \end{aligned}$$

for all \(t\in [-T,T]\) and \(\vert x\vert > \xi \). Set \(\zeta =\max \{\vert W(t,x)\vert \vert t \in [-T,T], \vert x\vert \leq \xi \}\), hence we have

$$\begin{aligned} W(t,x)\geq \biggl( \frac{\pi^{2}}{2T^{2}}+\varepsilon_{1} \biggr) \bigl( \vert x\vert ^{2}- \xi^{2}\bigr)-\zeta. \end{aligned}$$

Set

$$\begin{aligned} Q_{1}(t)=\textstyle\begin{cases} \sin(\omega t)e,&t\in [-T,T], \\ 0,&t\in R\setminus [-T,T], \end{cases}\displaystyle \end{aligned}$$

where \(\omega =\frac{\pi }{T}\), \(e=(1,0,\ldots,0)\). It can be easily checked that \(( \frac{\pi^{2}}{2T^{2}}+\varepsilon_{1} ) m>M\), where

$$\begin{aligned} M=\frac{1}{2} \int^{T}_{-T}\bigl\vert \dot{Q}_{1}(t) \bigr\vert ^{2}\,dt,\qquad m= \int^{T}_{-T}\bigl\vert Q_{1}(t)\bigr\vert ^{2}\,dt. \end{aligned}$$

By (4), for every \(r \in R\setminus \{0\}\), the following inequality holds:

$$\begin{aligned} I(rQ_{1}) =& \frac{1}{2} \int^{T}_{-T}\bigl\vert r\dot{Q}_{1}(t) \bigr\vert ^{2}\,dt- \int ^{T}_{-T}W\bigl(t,rQ_{1}(t)\bigr)\,dt+ \int^{T}_{-T}\bigl(f(t),rQ_{1}(t)\bigr)\,dt{ } \\ \leq & \frac{\vert r\vert ^{2}}{2} \int^{T}_{-T}\bigl\vert \dot{Q}_{2}(t) \bigr\vert ^{2}\,dt- \biggl( \frac{\pi^{2}}{2T^{2}}+\varepsilon_{1} \biggr) \vert r\vert ^{2} \int^{T} _{-T}\bigl\vert Q_{1}(t)\bigr\vert ^{2}\,dt{ } \\ & {} +\vert r\vert \delta m^{1/2}+2T \biggl( \biggl( \frac{\pi^{2}}{2T^{2}}+ \varepsilon_{1} \biggr) \xi^{2}+\zeta \biggr) \\ =& - \biggl( \biggl( \frac{\pi^{2}}{2T^{2}}+\varepsilon_{1} \biggr) m-M \biggr) \vert r\vert ^{2}+\vert r\vert \delta m^{1/2} +2T \biggl( \biggl( \frac{\pi^{2}}{2T ^{2}}+\varepsilon_{1} \biggr) \xi^{2}+\zeta \biggr), \end{aligned}$$

which implies that there exists \(r \in R \setminus \{ 0\} \) such that \(\Vert r Q_{1}\Vert > \varrho \) and \(I(r Q_{1}) < 0\). Set \(e(t)=r Q_{1}(t)\). Then \(\Vert e\Vert > \varrho \) and \(I(e) < 0\). □

Lemma 2.3

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 \(\Vert I^{\prime }(u_{n})\Vert (1+\Vert u_{n}\Vert ) \rightarrow 0\) as \(n \rightarrow \infty \). Then there exists a constant \(M_{1}> 0\) such that

$$\begin{aligned} \bigl\vert I(u_{n})\bigr\vert \leq M_{1},\qquad \bigl\Vert I^{\prime }(u_{n})\bigr\Vert \bigl(1+\Vert u_{n} \Vert \bigr) \leq M_{1}. \end{aligned}$$
(8)

Now we prove that \(\{u_{n}\}\) is bounded in E. Arguing in an indirect way, we assume that \(\Vert u_{n}\Vert \rightarrow +\infty \) as \(n\rightarrow \infty \). Set \(z_{n}=\frac{u_{n}}{\Vert u_{n}\Vert }\), then \(\Vert z_{n}\Vert =1\), which implies that there exists a subsequence of \(\{z_{n}\}\), still denoted by \(\{z_{n}\}\), such that \(z_{n}\rightharpoonup z_{0}\) in E. By (4) and (8), we get

$$\begin{aligned} \biggl\vert \int_{ R}\frac{W(t,u_{n}(t))}{\Vert u_{n}\Vert ^{2}}\,dt-\frac{1}{2}\biggr\vert =& \biggl\vert -\frac{I(u_{n})}{\Vert u_{n}\Vert ^{2}}+ \int_{ R}\frac{(f(t),u_{n}(t))}{ \Vert u_{n}\Vert ^{2}}\,dt\biggr\vert \\ \leq &\frac{M_{1}+\delta \Vert u_{n}\Vert }{\Vert u_{n}\Vert ^{2}}, \end{aligned}$$
(9)

which implies that

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

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

Case 1: \(z_{0}\equiv 0\). Let \(s>\frac{\delta C_{1}}{2}\). From \((W1)\) and \((W4)\), we can deduce that there exists \(M_{2}>0\) such that

$$\begin{aligned} \bigl\vert W(t,x)\bigr\vert \leq M_{2}\bigl(\vert x\vert ^{2}+\vert x\vert ^{\tau }\bigr) \quad \forall t \in R, \end{aligned}$$
(11)

and

$$\begin{aligned} \bigl\vert \bigl(\nabla W(t,x),x\bigr)\bigr\vert \leq M_{2}\bigl( \vert x\vert ^{2}+\vert x\vert ^{\tau }\bigr) \quad \forall t \in R. \end{aligned}$$
(12)

By the compactness of the embedding, one can obtain

$$\begin{aligned} \limsup_{n\rightarrow \infty } \int_{ R}\bigl\vert W\bigl(t,sz_{n}(t)\bigr)\bigr\vert \,dt\leq M_{2} \limsup_{n\rightarrow \infty } \int_{ R}\bigl( s^{2}\vert z_{n}\vert ^{2}+s^{\tau }\vert z_{n}\vert ^{\tau } \bigr)\,dt=0. \end{aligned}$$
(13)

Set \(\lambda_{n}=\frac{s}{\Vert u_{n}\Vert }\). It follows from (8)–(13), \((W2)\), and \((W5)\) that

$$\begin{aligned} M_{1} \geq & I(u_{n}) \\ =&I(\lambda_{n}u_{n})+\frac{1-\lambda_{n}^{2}}{2}\Vert u_{n}\Vert ^{2} \\ &{}+ \int _{ R} \bigl( W\bigl(t,\lambda_{n}u_{n}(t) \bigr)-W\bigl(t,u_{n}(t)\bigr) \bigr)\,dt+(1-\lambda _{n}) \int_{ R}\bigl(f(t),u_{n}(t)\bigr)\,dt \\ =&I(\lambda_{n}u_{n})+ \biggl( \frac{1-\lambda_{n}^{2}}{2}- \lambda_{n} \biggr) \bigl\langle I'(u_{n}),u_{n} \bigr\rangle \\ &{}+ \biggl( \lambda_{n}\Vert u_{n}\Vert ^{2}+ \biggl( \frac{(1- \lambda_{n})^{2}}{2}+\lambda_{n} \biggr) \int_{ R}\bigl(f(t),u_{n}(t)\bigr)\,dt \biggr) \\ &+ \int_{ R} \biggl( \biggl( \frac{1-\lambda_{n}^{2}}{2}- \lambda_{n} \biggr) \bigl( \nabla W\bigl(t,u_{n}(t) \bigr),u_{n}(t)\bigr)+W\bigl(t,\lambda_{n}u_{n}(t) \bigr)-W\bigl(t,u_{n}(t)\bigr) \biggr)\,dt \\ \geq & I(sz_{n})+o(1)+ \biggl( s-\delta C_{1} \biggl( \frac{(1-\lambda_{n})^{2}}{2}+ \lambda_{n} \biggr) \biggr) \Vert u_{n}\Vert \\ &{}+ \int_{\vert u_{n}\vert \geq r_{\infty }} \biggl( \biggl( \frac{1-\lambda_{n}^{2}}{2}- \lambda_{n} \biggr) \bigl(\nabla W\bigl(t,u_{n}(t) \bigr),u_{n}(t)\bigr) \\ &{}+W\bigl(t,\lambda_{n}u _{n}(t)\bigr)-W\bigl(t,u_{n}(t)\bigr) \biggr)\,dt{ } \\ &{}+ \int_{\vert u_{n}\vert \leq r_{\infty }} \biggl( \biggl( \frac{1-\lambda_{n}^{2}}{2}- \lambda_{n} \biggr) \bigl(\nabla W\bigl(t,u_{n}(t) \bigr),u_{n}(t)\bigr) \\ &{}+W\bigl(t,\lambda_{n}u _{n}(t)\bigr)-W\bigl(t,u_{n}(t)\bigr) \biggr)\,dt \\ \geq & \frac{s^{2}}{2}- \int_{ R}W\bigl(t,sz_{n}(t)\bigr)\,dt+ \int_{ R}\bigl(f(t),sz _{n}(t)\bigr)\,dt+o(1) \\ &{}-d_{2} \int_{\vert u_{n}\vert \geq r_{\infty }}\lambda_{n}^{\mu }\bigl\vert u_{n}(t)\bigr\vert ^{ \mu }\,dt+ \int_{\vert u_{n}\vert \leq r_{\infty }} \biggl( -\frac{\lambda_{n}^{2}+ \lambda_{n}}{2}\bigl(\nabla W \bigl(t,u_{n}(t)\bigr),u_{n}(t)\bigr) \biggr)\,dt \\ \geq & \frac{s^{2}}{2}-\delta C_{1} \int_{ R}\bigl\vert z_{n}(t)\bigr\vert \,dt-d _{2}s^{\mu } \int_{\vert u_{n}\vert \geq r_{\infty }}\bigl\vert z_{n}(t)\bigr\vert ^{\mu }\,dt+o(1) \\ &{}+M_{2} \int_{\vert u_{n}\vert \leq r_{\infty }} \biggl( -\frac{\lambda_{n}^{2}+ \lambda_{n}}{2}\bigl(\bigl\vert u_{n}(t)\bigr\vert ^{2}+\bigl\vert u_{n}(t) \bigr\vert ^{\tau }\bigr) \biggr)\,dt \\ \geq & \frac{s^{2}}{2}-\frac{M_{2}}{2} \int_{\vert u_{n}\vert \leq r_{\infty }} \bigl( \lambda_{n}^{2}\bigl\vert u_{n}(t)\bigr\vert ^{2}+ \lambda_{n} \bigl\vert u_{n}(t)\bigr\vert ^{2}+\lambda_{n}^{2} \bigl\vert u_{n}(t)\bigr\vert ^{\tau }+\lambda _{n}\bigl\vert u_{n}(t)\bigr\vert ^{\tau } \bigr)\,dt+o(1) \\ \geq & \frac{s^{2}}{2}-\frac{M_{2}}{2} \int_{\vert u_{n}\vert \leq r_{\infty }} \bigl( s^{2}\bigl\vert z_{n}(t)\bigr\vert ^{2}+r_{\infty }s\bigl\vert z_{n}(t)\bigr\vert +r_{\infty }^{\tau -2}s^{2} \bigl\vert z_{n}(t)\bigr\vert ^{2}+r_{\infty }^{\tau -1}s \bigl\vert z_{n}(t)\bigr\vert \bigr)\,dt+o(1) \\ \geq &\frac{s^{2}}{2}-o(1) \end{aligned}$$

for s and n large enough, which is a contradiction. Hence \(\Vert u_{n}\Vert \) is still bounded in this case, which implies that \(\{u_{n}\}\) is bounded in E.

Case 2: \(z_{0}\not \equiv 0\). Let \(\Omega =\{t\in R\vert \vert z_{0}(t)\vert >0 \}\). Then we can see that \(\operatorname{meas}(\Omega)>0\), where meas denotes the Lebesgue measure. Since \(\Vert u_{n}\Vert \rightarrow +\infty \) as \(n\rightarrow \infty \) and \(\vert u_{n}(t)\vert =\vert z_{n}(t)\vert \cdot \Vert u_{n}\Vert \), then we have \(\vert u_{n}(t)\vert \rightarrow +\infty \) as \(n\rightarrow \infty \) for a.e. \(t\in \Omega \). By \((W2)\), \((W3)\), and Fatou’s lemma, we can obtain

$$\begin{aligned} \liminf_{n\rightarrow \infty } \int_{ R}\frac{W(t,u_{n}(t))}{\Vert u_{n}\Vert ^{2}}\,dt \geq &\liminf _{n\rightarrow \infty } \int_{\Omega }\frac{W(t,u _{n}(t))}{\Vert u_{n}\Vert ^{2}}\,dt{ } \\ =& \liminf_{n\rightarrow \infty } \int_{\Omega }\frac{W(t,u _{n}(t))}{\vert u_{n}\vert ^{2}}\bigl\vert z_{n}(t) \bigr\vert ^{2}\,dt \\ =& +\infty, \end{aligned}$$

which contradicts (10). So \(\Vert u_{n}\Vert \) is bounded in this case. □

By a standard argument, we see that \(\{u_{n}\}\) has a convergent subsequence in E. Hence I satisfies the \((C)\) condition.

Proof of Theorem 1.1

The proof of this theorem is divided into two steps.

Step 1: We show that there exists a function \(u_{0}\in E\) such that \(I'(u_{0})=0\) and \(I(u_{0})<0\). Let \(f(t)=(f_{1}(t), f_{2}(t),\ldots,f_{N}(t))\), where \(f_{i}(t)\in C( R, R)\) (\(i=1,2,\ldots,N\)). Since \(f\not \equiv 0\), there exists \(i_{0}\in [1,N]\cap Z\) such that \(f_{i_{0}}(t)\not \equiv 0\). Without loss of generalization, we assume that there exist an interval \((a,b)\subset R\) and a constant \(A>0\) such that

$$\begin{aligned} f_{i_{0}}(t)\geq A\quad \mbox{for all } t\in (a,b). \end{aligned}$$

We choose a function \(\psi_{0}\in C_{0}^{\infty }(a,b)\) satisfying

$$\begin{aligned} \textstyle\begin{cases} \psi_{0}(t)=-f_{i_{0}}(t),&t\in ( \frac{3a+b}{4}, \frac{a+3b}{4} ), \\ \psi_{0}(t)\leq 0,&t\in ( a,b) \setminus ( \frac{3a+b}{4}, \frac{a+3b}{4} ), \\ \vert \psi_{0}'(t)\vert \leq 2,&t\in ( a,b). \end{cases}\displaystyle \end{aligned}$$

Set \(\psi (t)=(\psi_{1}(t),\psi_{2}(t),\ldots,\psi_{N}(t))\), where \(\psi_{j}(t)=0\) for all \(j\in [1,N]\cap Z\setminus \{i_{0}\}\) and \(\psi_{i}(t)=\psi_{0}(t)\) for \(j=i_{0}\). Therefore, \(\psi \in E\) and we can deduce that

$$\begin{aligned} \int_{ R}\bigl(f(t),\psi (t)\bigr)\,dt= \int_{a}^{b}f_{i_{0}}(t) \psi_{0}(t)\,dt \leq - \int_{\frac{3a+b}{4}}^{\frac{a+3b}{4}}f_{i_{0}}^{2}(t)\,dt\leq -\frac{A ^{2}}{2}(b-a)< 0. \end{aligned}$$

Hence we have

$$\begin{aligned} I(r\psi) =& \frac{1}{2} \int_{ R}\bigl\vert r\dot{\psi }(t)\bigr\vert ^{2}\,dt- \int_{ R}W\bigl(t,r \psi (t)\bigr)\,dt+ \int_{ R}\bigl(f(t),r\psi (t)\bigr)\,dt{ } \\ \leq & \frac{r^{2}}{2} \int_{ R}\bigl\vert \dot{\psi }(t)\bigr\vert ^{2}\,dt-M_{2} \biggl( r^{2} \int_{ R}\bigl\vert \psi (t)\bigr\vert ^{2}\,dt+r^{\tau } \int_{ R}\bigl\vert \psi (t)\bigr\vert ^{ \tau } \biggr)\,dt+r \int_{ R}\bigl(f(t),\psi (t)\bigr)\,dt \\ < &0 \end{aligned}$$

for \(r>0\) small enough. Then we obtain

$$\begin{aligned} c_{0}=\inf \bigl\{ I(u): u\in B_{\varrho }\bigr\} < 0, \end{aligned}$$

where ϱ is defined in Lemma 2.1 and \(B_{\varrho }=\{u\in E\vert \Vert u\Vert \leq \varrho \}\). By Ekeland’s variational principle, there exists a sequence \(\{u_{n}\}\subset B_{\varrho }\) such that

$$\begin{aligned} c_{0}\leq I(u_{n})\leq c_{0}+ \frac{1}{n}, \end{aligned}$$

and

$$\begin{aligned} I(w)\leq I(u_{n})-\frac{1}{n}\Vert w-u_{n}\Vert \end{aligned}$$

for all \(w\in B_{\varrho }\). Then, by a standard procedure, we can show that \(\{u_{n}\}\) is a \((C)\) sequence of I. Therefore, it follows from Lemma 2.3 that there exists a function \(u_{0}\in E\) such that \(I'(u_{0})=0\) and \(I(u_{0})<0\).

Step 2: By the mountain pass theorem and Lemmas 2.1–2.3, there exists \({\widetilde{u}_{0}}\in E\) such that \(I'(\widetilde{u}_{0})=0\) and \(I(\widetilde{u}_{0})>0\).

Then we finish the proof of Theorem 1.1. □