Set \(Y=Y_{k_{0}}\), \(Z=Z_{k_{0}}\).
Lemma 3.1
Suppose that (W1) and (W2) are satisfied. Then Ψ is bounded from below on Z.
Proof
By virtue of (W1), (W2), (2.3), (2.5), and (2.8), we have
$$\begin{aligned} \Psi(u) =&\frac{1}{2}\bigl\Vert u^{+}\bigr\Vert ^{2}-\frac{1}{2}\bigl\Vert u^{-}\bigr\Vert ^{2}-\sum _{n\in \mathbb{Z}}W\bigl(n,u(n)\bigr) \\ =&\frac{1}{2}\Vert u\Vert ^{2}-\sum _{n\in\mathbb{Z}}W\bigl(n,u(n)\bigr) \\ \geq&\frac{1}{2}\Vert u\Vert ^{2}-m_{0}\Vert u \Vert _{2}^{2}-\bar{a}\Vert u\Vert _{1}- \bar{b}\Vert u\Vert _{\nu+1}^{\nu+1} \\ \geq&\frac{1}{4}\Vert u\Vert ^{2}-\bar{a}D_{1} \Vert u\Vert -\bar{b} D_{\nu+1}^{\nu +1}\Vert u\Vert ^{\nu+1}\rightarrow+\infty \end{aligned}$$
(3.1)
as \(\|u\|\rightarrow\infty\) and \(u\in Z_{k_{0}}\). The proof is completed. □
Lemma 3.2
Assume that (H4) holds. Then for each finite dimensional subspace
\(\widetilde{E}\subset E\), there are positive constants
\(\rho=\rho(\widetilde{E})\)
and
\(\alpha=\alpha(\widetilde{E})\)
such that
\(\Psi|_{B_{\rho}\cap \widetilde{E}}\leq0\)
and
\(\Psi|_{\partial B_{\rho}\cap\widetilde {E}}\leq-\alpha\), where
\(B_{\rho}=\{x\in E:\|x\|\leq\rho\}\).
Proof
Let \(\widetilde{E}\subset E\) be any finite dimensional subspace. Then there exists \(M_{0}>0\) such that
$$ \|u\|^{2}\leq M_{0}\|u\|_{2}^{2}, \quad \forall u\in\widetilde{E}. $$
(3.2)
By virtue of (H4), for \(M_{0}\) given above, there exists a constant \(\sigma>0\),
$$ W(t,u)\geq M_{0}|u|^{2}, \quad \forall t\in \mathbb{R} \mbox{ and } |u|\leq \sigma. $$
(3.3)
In view of (2.8), for any \(u\in\widetilde{E}\) with \(\| u\|\leq\frac{\sigma}{D_{\infty}}\), we have
$$ \|u\|_{\infty}\leq\sigma. $$
(3.4)
By (3.2)-(3.4), we have
$$\begin{aligned} \Psi(u) =&\frac{1}{2}\bigl\Vert u^{+}\bigr\Vert ^{2}-\frac{1}{2}\bigl\Vert u^{-}\bigr\Vert ^{2}+ \int_{\mathbb {R}}W\bigl(t,u(t)\bigr)\,dt \\ \leq&\frac{1}{2}\bigl\Vert u^{+}\bigr\Vert ^{2}- \int_{\mathbb{R}}W\bigl(t,u(t)\bigr)\,dt \\ \leq&\frac{1}{2}\bigl\Vert u^{+}\bigr\Vert ^{2}-M_{0} \Vert u\Vert _{2}^{2} \\ \leq&-\frac{1}{2}\Vert u\Vert ^{2} \end{aligned}$$
(3.5)
for any \(u=u^{-}+u^{0}+u^{+}\in\widetilde{E}\) with \(\|u\|\leq\frac{\sigma }{D_{\infty}}\). Then there exist \(\rho=\rho(\widetilde{E})>0\) and \(\alpha=\alpha(\widetilde{E})>0\) such that
$$ \Psi(u)\leq0,\quad \forall u\in B_{\rho}\cap\widetilde{E}; \qquad \Psi (u)\leq-\alpha, \quad \forall u\in\partial B_{\rho}\cap\widetilde{E}. $$
The proof is completed. □
Lemma 3.3
Under the assumptions of Theorem
1.3, Ψ satisfies condition
\((C)\).
Proof
Let \(\{u_{n}\}\subset E\) is a \((C)\) sequence of Ψ, that is, \(\{\Psi(u_{n})\}\) is bounded and
$$ \bigl(1+\Vert u_{n}\Vert \bigr)\bigl\Vert \Psi^{\prime}(u_{n})\bigr\Vert \rightarrow0 \quad \mbox{as } n \rightarrow\infty, $$
(3.6)
then there exists a constant \(M_{1}>0\) such that
$$ \bigl\vert \Phi(u_{m})\bigr\vert \leq M_{1}, \qquad \bigl(1+\Vert u_{n}\Vert \bigr)\bigl\Vert \Psi^{\prime}(u_{n})\bigr\Vert \leq M_{1} $$
(3.7)
for every \(n\in\mathbb{N}\). We choose \(k\geq n^{+}+1\) large enough such that
$$ \|u\|^{2}\geq2m_{0}\|u\|_{2}^{2}, \quad \forall u\in Z_{k}, $$
(3.8)
where \(m_{0}=\sup_{t\in\mathbb{R}}[\sup_{x\in\mathbb {R}^{N},|x|=1}(S(t)x,x)]\). We now prove that \(\{u_{n}\}\) is bounded in E. In fact, if not, we may assume by contradiction that \(\|u_{n}\| \rightarrow\infty\) as \(n\rightarrow\infty\). Let \(u_{n}=\tilde {w}_{n}+\tilde{v}_{n}\), \(z_{n}=\frac{u_{n}}{\|u_{n}\|}\), then \(\|z_{n}\|=1\), \(z_{n}=w_{n}+v_{n}\in E\), where \(w_{n}=\frac{\tilde{w}_{n}}{\|u_{n}\|}\), \(v_{n}=\frac{\tilde{v}_{n}}{\|u_{n}\|}\), \(\tilde{w}_{n}\in Y_{k}\), \(\tilde {v}_{n}\in Z_{k}\). After passing to a subsequence, we have \(z_{n}\rightharpoonup z\), \(w_{n}\rightarrow w\), and \(\gamma=\lim_{n\rightarrow\infty}\|v_{n}\|\) exists.
Case 1. \(\gamma=0\). Since \(\dim Y_{k}<\infty\), we obtain \(\| w_{n}\|\rightarrow\|w\|=1\). It follows from (3.7) that
$$ \frac{3}{2}M_{1}\geq\Psi(u_{n})- \frac{1}{2}\Psi^{\prime}(u_{n})u_{n} \geq \int_{\mathbb{R}} \biggl(W\bigl(t,u_{n}(t)\bigr)- \frac{1}{2}\bigl(\nabla W\bigl(t,u_{n}(t)\bigr),u_{n}(t) \bigr) \biggr)\, dt. $$
(3.9)
By virtue of (H6), for any \(\eta>0\), there exists \(M_{2}>0\) such that
$$ \widetilde{W}(t,u)=W(t,u)-\frac{1}{2}\bigl(\nabla W(t,u),u \bigr)\geq\eta, \quad \forall t\in\mathbb{R}, |u|\geq M_{2}. $$
(3.10)
For any \(\varepsilon>0\), define \(\Lambda_{\varepsilon}:=\{t\in\mathbb {R}:|w(t)|\geq\varepsilon\}\) and \(\Lambda_{n\varepsilon}:=\{t\in \mathbb{R}:|v_{n}(t)|\geq\frac{\varepsilon}{2}\}\). First, we claim that there exists \(\varepsilon_{0}>0\) such that
$$ \operatorname{meas}\bigl\{ t\in\mathbb{R}:\bigl|u(t)\bigr|\geq\varepsilon_{0} \bigr\} \geq \varepsilon_{0}, \quad \forall u\in Y_{k} \mbox{ with } \|u\|=1. $$
Otherwise, for any positive integer m, there exists \(w_{m}\in Y_{k}\) with \(\|w_{m}\|=1\) such that
$$ \operatorname{meas} \biggl\{ t\in\mathbb{R}:\bigl|w_{m}(t)\bigr| \geq\frac{1}{m} \biggr\} < \frac{1}{m}. $$
(3.11)
Passing to a subsequence if necessary, we may assume \(w_{m}\rightarrow w_{0}\) in E for some \(w_{0}\in Y_{k}\) since \(\dim Y_{k}<\infty\). Evidently, \(\|w_{0}\|=1\). By the equivalence of the norms on the finite dimensional space \(Y_{k}\), we have
$$ \int_{\mathbb{R}}\bigl\vert w_{m}(t)-w_{0}(t) \bigr\vert ^{2}\, dt\rightarrow0\quad \mbox{as } m\rightarrow\infty. $$
(3.12)
Thus there exists \(\varepsilon_{1}>0\) such that
$$ \operatorname{meas} \bigl\{ t\in\mathbb{R}:\bigl\vert w_{0}(t)\bigr\vert \geq\varepsilon_{1} \bigr\} \geq \varepsilon_{1}. $$
(3.13)
In fact, if not, then, for all positive integers m, we have
$$ \operatorname{meas} \biggl\{ t\in\mathbb{R}:\bigl\vert w_{0}(t)\bigr\vert \geq\frac{1}{m} \biggr\} =0. $$
(3.14)
It implies that
$$ 0\leq \int_{\mathbb{R}}\bigl\vert w_{0}(t)\bigr\vert ^{4}\, dt< \frac{1}{m^{2}}\|w_{0}\| _{2}^{2} \rightarrow0 \quad \mbox{as } m\rightarrow\infty. $$
Hence \(w_{0}=0\), which contradicts that \(\|w_{0}\|_{0}=1\). Thus, (3.13) holds.
Now set
$$ \Upsilon_{0}= \bigl\{ t\in\mathbb{R}:\bigl\vert w_{0}(t) \bigr\vert \geq\varepsilon_{1} \bigr\} , \qquad \Upsilon_{m}= \biggl\{ t\in\mathbb{R}:\bigl\vert w_{m}(t)\bigr\vert < \frac{1}{m} \biggr\} , $$
and \(\Upsilon_{m}^{c}=\mathbb{R}\setminus\Upsilon_{m}=\{t\in\mathbb {R}:|w_{m}(t)|\geq\frac{1}{m}\}\). By virtue of (3.11) and (3.13), we obtain
$$\begin{aligned} \begin{aligned} \operatorname{meas}(\Upsilon_{m}\cap\Upsilon_{0})&= \operatorname{meas}\bigl(\Upsilon _{0}\setminus\Upsilon_{m}^{c} \cap\Upsilon_{0}\bigr) \\ &\geq\operatorname{meas}(\Upsilon_{0})-\operatorname{meas}\bigl( \Upsilon_{m}^{c}\cap\Upsilon _{0}\bigr) \\ &\geq\varepsilon_{1}-\frac{1}{m} \end{aligned} \end{aligned}$$
for all positive integers m. Let m be large enough such that \(\frac {1}{2}\varepsilon_{1}-\frac{1}{m}>\frac{1}{4}\varepsilon_{1}\). Thus, for m large enough,
$$\begin{aligned} \int_{\mathbb{R}}\bigl\vert w_{m}(t)-w_{0}(t) \bigr\vert ^{2}\, dt \geq& \int_{t\in\Upsilon _{0}\cap\Upsilon_{m}}\bigl\vert w_{m}(t)-w_{0}(t) \bigr\vert ^{2}\, dt \\ \geq&\frac{1}{2} \int_{t\in\Upsilon_{0}\cap\Upsilon _{m}}\bigl\vert w_{0}(t)\bigr\vert ^{2}\, dt- \int_{t\in\Upsilon_{0}\cap\Upsilon_{m}}\bigl\vert w_{m}(t)\bigr\vert ^{2}\, dt \\ \geq& \biggl(\frac{1}{2}\varepsilon_{1}-\frac{1}{m} \biggr) \operatorname{meas}(\Upsilon_{m}\cap\Upsilon_{0}) \\ \geq&\frac{\varepsilon_{1}^{2}}{16}>0, \end{aligned}$$
which is a contradiction to (3.12). Thus, there exists \(\varepsilon>0\) such that \(\operatorname{meas}(\Lambda_{\varepsilon})\geq \varepsilon\).
In view of (2.5), we obtain
$$ \operatorname{meas}(\Lambda_{n\varepsilon})\leq\frac{4}{\varepsilon^{2}} \int _{\mathbb{R}}\bigl\vert v_{n}(t)\bigr\vert ^{2}\, dt \leq\frac{4D_{2}^{2}}{\varepsilon^{2}}\|v_{n}\|^{2} \rightarrow0\quad \mbox{as } n\rightarrow\infty. $$
Then we have \(\operatorname{meas}(\Lambda_{\varepsilon}\setminus\Lambda _{n\varepsilon})\rightarrow\operatorname{meas}(\Lambda_{\varepsilon})\) as \(n\rightarrow\infty\). Therefore, there exists \(N_{0}>0\) such that \(|z_{n}(t)|\geq\frac{\varepsilon}{3}\), \(\forall t\in\Lambda _{\varepsilon}\setminus\Lambda_{n\varepsilon}\) and \(n\geq N_{0}\), then we have \(|u_{n}(t)|\geq\frac{\varepsilon}{3}\|u_{n}\|\), \(\forall t\in \Lambda_{\varepsilon}\setminus\Lambda_{n\varepsilon}\) and \(n\geq N_{0}\). By (W3), (3.9), and (3.10), there exists \(N_{1}>0\) such that
$$ \frac{3}{2}M_{1}\geq \int_{\mathbb{R}}\widetilde{W}\bigl(t,u_{n}(t)\bigr)\, dt \geq \int_{t\in\Lambda_{\varepsilon}\setminus\Lambda_{n\varepsilon }}\eta \, dt\geq\eta\operatorname{meas}( \Lambda_{\varepsilon}\setminus\Lambda _{n\varepsilon}), \quad \forall n\geq N_{1}, $$
which gives a contradiction due to the arbitrariness of η.
Case 2. \(\gamma>0\). In view of (W1), (W2), (2.5), (3.7), (3.8), and Hölder’s inequality, we get
$$\begin{aligned} M_{1} \geq&\Psi^{\prime}(u_{n}) \tilde{v}_{n}=\|\tilde{v}_{n}\|^{2}- \int _{\mathbb{R}}\bigl(\nabla W\bigl(t,u_{n}(t)\bigr), \tilde{v}_{n}(t)\bigr)\,dt \\ \geq&\|\tilde{v}_{n}\|^{2}- \int_{\mathbb{R}} \bigl[\bigl(S(t)u_{n}(t), \tilde{v}_{n}(t)\bigr) + \bigl(\bar{a}+\bar{b}\bigl\vert u_{n}(t)\bigr\vert ^{\nu} \bigr)\bigl\vert \bar{v}_{n}(t)\bigr\vert \bigr]\,dt \\ \geq&\|\tilde{v}_{n}\|^{2}-m_{0}\| \tilde{v}_{n}\|_{2}^{2}-\bar{a}\|\tilde {v}_{n}\|_{1} -\bar{b}\|u_{n} \|_{2\nu}^{\nu}\|\tilde{v}_{n}\|_{2} \\ \geq&\frac{1}{2}\|\tilde{v}_{n}\|^{2}- \bar{a}D_{1}\|\tilde{v}_{n}\|- \bar{b} D_{2}D_{2\nu}^{\nu} \|u_{n}\|^{\nu}\|\tilde{v}_{n}\| \\ \geq&\frac{1}{2}\|\tilde{v}_{n}\|^{2}- \bar{a}D_{1}\|u_{n}\|- \bar{b} D_{2}D_{2\nu}^{\nu} \|u_{n}\|^{\nu+1}. \end{aligned}$$
(3.15)
Dividing by \(\|u_{n}\|^{2}\) on both sides of (3.15), we obtain
$$ 0\geq\frac{\gamma^{2}}{2}>0, $$
which gives a contradiction.
Thus, \(\{u_{n}\}\) is bounded. Next, we show that \(\{u_{n}\}\) has a convergent subsequence. In view of the boundedness of \(\{u_{n}\}\), without loss of generality, we may assume that
$$ u_{n}\rightharpoonup u,\qquad u_{n}^{+} \rightharpoonup u^{+}, \qquad u_{n}^{-}\rightarrow u^{-},\qquad u_{n}^{0}\rightarrow u^{0}. $$
(3.16)
It follows from (2.4) that
$$\begin{aligned} \bigl\Vert u_{n}^{+}-u^{+}\bigr\Vert ^{2} =& \bigl(\Psi^{\prime}(u_{n})-\Psi^{\prime}(u)\bigr) \bigl(u_{n}^{+}-u^{+}\bigr) \\ &{}+ \int_{\mathbb{R}}\bigl(\nabla W(t,u_{n})-\nabla W(t,u),u_{n}^{+}-u^{+}\bigr)\, dt. \end{aligned}$$
(3.17)
It is clear that
$$ \bigl(\Psi^{\prime}(u_{n})-\Psi^{\prime}(u) \bigr) \bigl(u_{n}^{+}-u^{+}\bigr)\rightarrow0 \quad \mbox{as } n \rightarrow\infty. $$
(3.18)
By (2.5), Lemma 2.1, (W1), (W2), and Hölder’s inequality, we have
$$\begin{aligned}& \int_{\mathbb{R}}\bigl(\nabla W(t,u_{n})-\nabla W(t,u),u_{n}^{+}-u^{+}\bigr)\, dt \\& \quad \leq \bar{a}\bigl\Vert u_{n}^{+}-u^{+}\bigr\Vert _{1}+ \bigl(m_{0}\Vert u_{n}\Vert _{2}+\bar{b}\Vert u_{n}\Vert _{2\nu}^{\nu} \bigr)\bigl\Vert u_{n}^{+}-u^{+}\bigr\Vert _{2} \\& \qquad {}+\bar{a}\bigl\Vert u_{n}^{+}-u^{+}\bigr\Vert _{1}+ \bigl(m_{0}\Vert u\Vert _{2}+\bar{b}\Vert u\Vert _{2\nu }^{\nu} \bigr)\bigl\Vert u_{n}^{+}-u^{+}\bigr\Vert _{2} \\& \quad \leq 2\bar{a}\bigl\Vert u_{n}^{+}-u^{+}\bigr\Vert _{1}+ \bigl(m_{0}D_{2}\Vert u_{n} \Vert +\bar{b}D_{2\nu }^{\nu} \Vert u_{n}\Vert ^{\nu} \bigr)\bigl\Vert u_{n}^{+}-u^{+}\bigr\Vert _{2} \\& \qquad {}+ \bigl(m_{0}D_{2}\Vert u\Vert + \bar{b}D_{2\nu}^{\nu} \Vert u\Vert ^{\nu} \bigr)\bigl\Vert u_{n}^{+}-u^{+}\bigr\Vert _{2}\rightarrow0 \end{aligned}$$
(3.19)
as \(n\rightarrow\infty\). Therefore, by (3.17)-(3.19), we get \(\|u_{m}^{+}-u^{+}\|\rightarrow0\) as \(n\rightarrow \infty\). Consequently, \(u_{n}\rightarrow u\). The proof is completed. □
Proof of Theorem 1.3
Obviously, \(\Psi\in C^{1}(X,\mathbb{R})\) is even and \(\Phi(0)=0\). It follows from Lemmas 3.1-3.3 that all conditions of Lemma 2.3 are satisfied. By Lemma 2.3, we see that Ψ possesses infinitely many nontrivial critical points, that is, system (1.1) possesses infinitely many nontrivial homoclinic orbits. □