Lemma 3.1
Assume that (\(\mathrm{L}_{\nu}\)) and (W1) hold. Then, for
\(u\in E\),
$$ \int_{\mathbb {R}}\bigl|W(t, u)\bigr|\,\mathrm{d}t\le \phi_{1}(T)\|u\|^{\gamma_{1}}+\phi _{2}(T)\| u \|^{\gamma_{2}}, \quad T\ge T_{0}, $$
(3.1)
where
$$\begin{aligned}& \phi_{1}(T)=a_{1}\beta^{\gamma_{1}} \biggl[ \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]^{-\gamma_{1}/(2-\gamma_{1})}\, \mathrm{d}t \biggr)^{1-\frac {\gamma_{1}}{2}} +\frac{K(\gamma_{1})}{T^{\kappa_{1}}} \biggr], \end{aligned}$$
(3.2)
$$\begin{aligned}& \phi_{2}(T)=a_{2}\beta^{\gamma_{2}} \biggl[ \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]^{-\gamma_{2}/(2-\gamma_{2})}\, \mathrm{d}t \biggr)^{1-\frac {\gamma_{2}}{2}} +\frac{K(\gamma_{2})}{T^{\kappa_{2}}} \biggr] \end{aligned}$$
(3.3)
and
$$ \kappa_{1}=\frac{(3-\nu)\gamma_{1}-2}{2}, \qquad \kappa_{2}=\frac {(3-\nu )\gamma_{2}-2}{2}. $$
(3.4)
Proof
For \(T\ge T_{0}\), it follows from (2.14), (2.17), (3.2), (3.3) and (W1) that
$$\begin{aligned} \int_{\mathbb {R}}\bigl|W(t, u)\bigr|\,\mathrm{d}t \le& a_{1}\int _{\mathbb {R}}\bigl|u(t)\bigr|^{\gamma_{1}}\,\mathrm{d}t+a_{2}\int _{\mathbb {R}}\bigl|u(t)\bigr|^{\gamma_{2}}\,\mathrm{d}t \\ \le& a_{1} \biggl[ \biggl(\int_{|t|\le T} \bigl[l(t)+l_{0}\bigr]^{-\gamma _{1}/(2-\gamma _{1})}\,\mathrm{d}t \biggr)^{1-\frac{\gamma_{1}}{2}} +\frac{K(\gamma_{1})}{T^{\kappa_{1}}} \biggr]\|u\|_{*}^{\gamma _{1}} \\ &{} +a_{2} \biggl[ \biggl(\int_{|t|\le T} \bigl[l(t)+l_{0}\bigr]^{-\gamma_{2}/(2-\gamma _{2})}\,\mathrm{d}t \biggr)^{1-\frac{\gamma_{2}}{2}} +\frac{K(\gamma_{2})}{T^{\kappa_{2}}} \biggr]\|u\|_{*}^{\gamma _{2}} \\ \le& a_{1}\beta^{\gamma_{1}} \biggl[ \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]^{-\gamma_{1}/(2-\gamma_{1})}\, \mathrm{d}t \biggr)^{1-\frac {\gamma_{1}}{2}} +\frac{K(\gamma_{1})}{T^{\kappa_{1}}} \biggr]\|u\|^{\gamma_{1}} \\ &{} +a_{2}\beta^{\gamma_{2}} \biggl[ \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]^{-\gamma _{2}/(2-\gamma_{2})}\, \mathrm{d}t \biggr)^{1-\frac{\gamma_{2}}{2}} +\frac{K(\gamma_{2})}{T^{\kappa_{2}}} \biggr]\|u\|^{\gamma_{2}} \\ = & \phi_{1}(T)\|u\|^{\gamma_{1}}+\phi_{2}(T)\|u \|^{\gamma_{2}}. \end{aligned}$$
□
Analogous to the proof of [17], Lemma 2.2, we can prove the following lemma.
Lemma 3.2
Assume that (\(\mathrm{L}_{\nu}\)), (W1) and (W2) hold. Then the functional
\(f: E\rightarrow{ \mathbb {R}}\)
defined by
$$ \Phi(u)=\frac{1}{2} \bigl(\bigl\| u^{+} \bigr\| ^{2}-\bigl\| u^{-}\bigr\| ^{2} \bigr)-\int _{\mathbb {R}}W(t, u)\,\mathrm{d}t $$
(3.5)
is well defined and of class
\(C^{1}(E, {\mathbb {R}})\)
and
$$ \bigl\langle \Phi'(u), v \bigr\rangle = \bigl\| u^{+}\bigr\| ^{2}-\bigl\| u^{-}\bigr\| ^{2} -\int _{\mathbb {R}}\bigl(\nabla W(t, u), v\bigr)\,\mathrm{d}t. $$
(3.6)
Furthermore, the critical points of Φ in
E
are classical solutions of (1.1) with
\(u(\pm\infty)=0\).
Proof of Theorem 1.1
In view of Lemma 3.2, \(\Phi\in C^{1}(E, \mathbb {R})\). In what follows, we divide the rest of the proof of Theorem 1.1 into four steps.
Step 1. Φ satisfies the (PS)-condition.
Assume that \(\{u_{n}\}_{n\in{\mathbb{N}}}\subset E\) is a (PS)-sequence: \(\{\Phi(u_{n})\}_{k\in{\mathbb{N}}}\) is bounded and \(\|\Phi'(u_{n})\| \rightarrow0\) as \(n\rightarrow+\infty\). In the sequel we write for any \(u\in E\)
$$ u^{1}(t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u(t) & \mbox{if } |u(t)|< 1,\\ 0 & \mbox{if } |u(t)| \ge1; \end{array}\displaystyle \displaystyle \right . \qquad u^{2}(t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} 0 & \mbox{if } |u(t)|< 1,\\ u(t) & \mbox{if } |u(t)| \ge1. \end{array}\displaystyle \displaystyle \displaystyle \displaystyle \displaystyle \right . $$
(3.7)
Then, by (3.5), (3.6), (3.7) and (W3), we get
$$\begin{aligned} \bigl\langle \Phi'(u_{n}), u_{n} \bigr\rangle -2\Phi(u_{n}) = & \int_{\mathbb {R}}\bigl[2W(t, u_{n})-\bigl(\nabla W(t, u_{n}), u_{n}\bigr) \bigr]\,\mathrm{d}t \\ \ge& b_{1}\int_{\mathbb {R}}\bigl|u^{1}_{n}\bigr|^{\gamma_{4}} \,\mathrm{d}t+b_{2}\int_{\mathbb {R}}\bigl|u^{2}_{n}\bigr|^{\gamma_{5}} \,\mathrm{d}t \\ = & b_{1}\bigl\| u^{1}_{n}\bigr\| _{\gamma_{4}}^{\gamma_{4}}+b_{2} \bigl\| u^{2}_{n}\bigr\| _{\gamma _{5}}^{\gamma_{5}}. \end{aligned}$$
It follows that there exists a constant \(C_{1}>0\) such that
$$ b_{1}\bigl\| u^{1}_{n} \bigr\| _{\gamma_{4}}^{\gamma_{4}}+b_{2}\bigl\| u^{2}_{n} \bigr\| _{\gamma _{5}}^{\gamma _{5}}\le C_{1}\bigl(1+\|u_{n}\|\bigr). $$
(3.8)
Since \(\dim(E^{-}\oplus E^{0})<+\infty\), there exists a constant \(C_{2}>0\) such that
$$\begin{aligned} \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}^{2} = & \bigl(u_{n}^{-}+u_{n}^{0}, u_{n} \bigr)_{2} \\ = & \bigl(u_{n}^{-}+u_{n}^{0}, u^{1}_{n} \bigr)_{2}+ \bigl(u_{n}^{-}+u_{n}^{0}, u^{2}_{n} \bigr)_{2} \\ \le& \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{\gamma_{4}'}\bigl\| u^{1}_{n}\bigr\| _{\gamma_{4}}+\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{\gamma_{5}'}\bigl\| u^{2}_{n}\bigr\| _{\gamma_{5}} \\ \le& C_{2}\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2} \bigl(\bigl\| u^{1}_{n}\bigr\| _{\gamma_{4}}+\bigl\| u^{2}_{n}\bigr\| _{\gamma_{5}} \bigr), \end{aligned}$$
(3.9)
where \(\gamma_{4}'=\gamma_{4}/(\gamma_{4}-1)\) and \(\gamma_{5}'=\gamma _{5}/(\gamma _{5}-1)\). Combining (3.8) with (3.9), one has
$$ \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}\le C_{3}\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}^{2}\le C_{4} \bigl(1+\| u_{n} \|^{2/\gamma_{4}}+\|u_{n}\|^{2/\gamma_{5}} \bigr). $$
(3.10)
Choose \(T_{2}>T_{0}\), it follows from (3.1) that
$$ \int_{\mathbb {R}}W(t, u_{n})\,\mathrm{d}t \le\phi_{1}(T_{2})\|u_{n}\|^{\gamma _{1}}+\phi _{2}(T_{2})\|u_{n}\|^{\gamma_{2}}. $$
(3.11)
From (3.5), (3.10) and (3.11), we obtain
$$\begin{aligned} \|u_{n}\|^{2} = & \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}+\bigl\| u_{n}^{+}\bigr\| ^{2} \\ = & \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}+2\Phi(u_{n})+\bigl\| u_{n}^{-} \bigr\| ^{2}+2\int_{\mathbb {R}}W(t, u_{n})\, \mathrm{d}t \\ \le& 2C_{4} \bigl(1+\|u_{n}\|^{2/\gamma_{4}}+ \|u_{n}\|^{2/\gamma_{5}} \bigr)+2\Phi(u_{n}) \\ &{} +2\phi_{1}(T_{2})\|u_{n}\|^{\gamma_{1}}+2 \phi_{2}(T_{2})\|u_{n}\|^{\gamma _{2}} \\ \le& C_{5} \bigl(1+\|u_{n}\|^{\gamma_{1}}+ \|u_{n}\|^{\gamma_{2}}+\|u_{n}\| ^{2/\gamma_{4}}+ \|u_{n}\|^{2/\gamma_{5}} \bigr). \end{aligned}$$
(3.12)
Since \(1<\gamma_{1}<\gamma_{2}<2\), \(1<\gamma_{5} \le\gamma_{4}<2\), it follows from (3.12) that \(\{\|u_{n}\|\}\) is bounded, and so \(\{\|u_{n}\|_{*}\}\) is bounded. Choose a constant \(\Lambda>0\) such that
$$ \|u_{n}\|_{\infty}\le\frac{1}{\sqrt{2}} \|u_{n}\|_{*}\le\Lambda, \quad n\in\mathbb{N}. $$
(3.13)
Passing to a subsequence if necessary, it can be assumed that \(u_{n}\rightharpoonup u_{0}\) in E. Hence \(u_{n}\rightarrow u_{0}\) in \(L^{\infty}_{\mathrm {\mathrm{loc}}}(\mathbb {R}, {\mathbb {R}}^{N})\); moreover, it is easy to verify that \(\{u_{n}(t)\}\) converges to \(u_{0}(t)\) point-wise for all \(t\in{\mathbb{R}}\). Hence, (3.13) yields that \(\|u_{0}\|_{\infty}\le\Lambda\). By (W2), there exists \(M_{3}>0\) such that
$$ \nabla W(t, x)\le M_{3}|x|^{\gamma_{3}-1}, \quad \forall x\in{ \mathbb {R}}^{N}, |x|\le\Lambda. $$
(3.14)
For any given number \(\varepsilon>0\), we can choose \(T_{3}>T_{0}\) such that
$$ \frac{K(\gamma_{3}) [ (\sqrt{2}\Lambda )^{\gamma _{3}}+\|u_{0}\| _{*}^{\gamma_{3}} ]}{T_{3}^{\kappa_{3}}}< \varepsilon. $$
(3.15)
Hence, from (2.16), (3.13), (3.14) and (3.15) we have that
$$\begin{aligned} \int_{|t|>T_{3}}\bigl|\nabla W(t, u_{n})- \nabla W(t, u_{0})\bigr||u_{n}-u_{0}|\,\mathrm{d}t \le& 2M_{3}\int_{|t|>T_{3}}\bigl(\bigl|u_{k}(t)\bigr|^{\gamma_{3}}+\bigl|u_{0}(t)\bigr|^{\gamma _{3}} \bigr)dt \\ \le& \frac{2M_{3}K(\gamma_{3})}{T_{3}^{\kappa_{3}}} \bigl(\|u_{k}\| _{*}^{\gamma _{3}}+ \|u_{0}\|_{*}^{\gamma_{3}} \bigr) \\ \le& \frac{2M_{3}K(\gamma_{3})}{T_{3}^{\kappa_{3}}} \bigl[ (\sqrt {2}\Lambda )^{\gamma_{3}}+ \|u_{0}\|_{*}^{\gamma_{3}} \bigr] \\ \le& 2M_{3}\varepsilon, \quad n\in{\mathbb{N}}. \end{aligned}$$
(3.16)
On the other hand, since \(u_{n}\rightarrow u_{0}\) in \(L^{\infty}_{\mathrm {\mathrm{loc}}}(\mathbb {R}, {\mathbb {R}}^{N})\), it follows from the continuity of \(\nabla W(t, x)\) that
$$ \int_{-T_{3}}^{T_{3}}\bigl|\nabla W(t, u_{n})-\nabla W(t, u_{0})\bigr||u_{n}-u_{0}| \,\mathrm{d}t = o(1). $$
(3.17)
Since ε is arbitrary, combining (3.16) with (3.17) we get
$$ \int_{\mathbb{R}}\bigl(\nabla W(t, u_{n})-\nabla W(t, u_{0}), u_{n}-u_{0} \bigr)\,\mathrm{d}t = o(1). $$
(3.18)
It follows from (3.6) that
$$\begin{aligned} \bigl\langle \Phi'(u_{n})- \Phi'(u_{0}), u_{n}-u_{0}\bigr\rangle = & \bigl\| u_{n}^{+}-u_{0}^{+} \bigr\| ^{2}-\bigl\| u_{n}^{-}-u_{0}^{-} \bigr\| ^{2} \\ &{} -\int_{{\mathbb{R}}}\bigl(\nabla W(t, u_{n})-\nabla W(t, u_{0}), u_{n}-u_{0}\bigr)\,\mathrm{d}t. \end{aligned}$$
(3.19)
Since \(\langle\Phi'(u_{n})-\Phi'(u_{0}), u_{n}-u_{0}\rangle=o(1)\), it follows from (3.18) and (3.19) that
$$ \bigl\| u_{n}^{+}-u_{0}^{+} \bigr\| ^{2}-\bigl\| u_{n}^{-}-u_{0}^{-} \bigr\| ^{2}=o(1). $$
(3.20)
Since \(u_{n}\rightharpoonup u_{0}\) in E and \(\dim(E^{-}\oplus E^{0})<+\infty\), it follows that
$$ \bigl\| u_{n}^{0}-u_{0}^{0} \bigr\| ^{2}+\bigl\| u_{n}^{-}-u_{0}^{-} \bigr\| ^{2}= o(1). $$
(3.21)
Combining (3.20) with (3.21), we have
$$ \|u_{n}-u_{0}\|^{2}=\bigl\| u_{n}^{+}-u_{0}^{+} \bigr\| ^{2}+\bigl\| u_{n}^{0}-u_{0}^{0} \bigr\| ^{2}+\bigl\| u_{n}^{-}-u_{0}^{-} \bigr\| ^{2}=o(1). $$
Hence, Φ satisfies the (PS)-condition.
Step 2. \(\Phi(u)\rightarrow+\infty\) as \(\|u\|\rightarrow+\infty\) and \(u\in E^{+}\).
It follows from (3.1) that
$$ \int_{\mathbb {R}}W(t, u)\,\mathrm{d}t \le \phi_{1}(T_{2})\|u\|^{\gamma_{1}}+\phi _{2}(T_{2}) \|u\|^{\gamma_{2}}, \quad \forall u\in E. $$
(3.22)
Hence, for \(u\in E^{+}\), it follows from (3.5) and (3.22) that
$$\begin{aligned} \Phi(u) = & \frac{1}{2}\|u\|^{2}-\int_{\mathbb {R}}W(t, u)\,\mathrm{d}t \\ \ge& \frac{1}{2}\|u\|^{2}-\phi_{1}(T_{2}) \|u\|^{\gamma_{1}} - \phi _{2}(T_{2})\| u\|^{\gamma_{2}} \rightarrow+\infty \end{aligned}$$
as \(\|u\|\rightarrow+\infty\) and \(u\in E^{+}\), since \(1<\gamma _{1}<\gamma_{2}<2\).
Step 3. Taking \(e\in E^{+}\) with \(\|e\|=1\), there exist \(s_{0}\in(0, 1)\) and \(\sigma_{0}>0\) such that
$$ \Phi(u)\le-\sigma_{0}, \quad \forall u\in S_{e}:=E^{-}\oplus E^{0}\oplus s_{0}e. $$
(3.23)
Set \(X=E^{-}\oplus E^{0}\oplus \mathbb {R}e\). For \(u=u^{-}+u^{0}+se\in X\), by (3.5), (3.7) and (W4),
$$\begin{aligned} \Phi(u) = & \frac{1}{2} \bigl(\|se\|^{2}- \bigl\| u^{-}\bigr\| ^{2} \bigr)-\int_{\mathbb {R}}W(t, u)\, \mathrm{d}t \\ \le& \frac{s^{2}}{2}-b_{3}\bigl\| u^{1} \bigr\| _{\gamma_{6}}^{\gamma_{6}}-b_{4}\bigl\| u^{2}\bigr\| _{\gamma_{7}}^{\gamma_{7}}. \end{aligned}$$
(3.24)
On the other hand, one sees that
$$\begin{aligned} s^{2}\|e\|_{2}^{2} = (se, se)_{2} = (se, u)_{2}=\bigl(se, u^{1}\bigr)_{2}+\bigl(se, u^{2}\bigr)_{2} \le |s| \bigl(\|e\|_{\gamma_{6}'}\bigl\| u^{1}\bigr\| _{\gamma_{6}}+\|e \|_{\gamma _{7}'}\bigl\| u^{2}\bigr\| _{\gamma_{7}} \bigr), \end{aligned}$$
where \(\gamma_{6}'=\gamma_{6}/(\gamma_{6}-1) > \gamma_{6}\) and \(\gamma _{7}'=\gamma _{7}/(\gamma_{7}-1) > \gamma_{7}\). Hence,
$$ s\le C_{6} \bigl(\bigl\| u^{1}\bigr\| _{\gamma_{6}}+ \min\bigl\{ \bigl\| u^{2}\bigr\| _{\gamma_{7}}, 1\bigr\} \bigr), \quad \forall s \in(0, 1). $$
(3.25)
Combining (3.24) with (3.25), we have
$$\begin{aligned} \Phi(u) \le& \frac{s^{2}}{2}-b_{3}\bigl\| u^{1} \bigr\| _{\gamma_{6}}^{\gamma _{6}}-b_{4}\bigl\| u^{2} \bigr\| _{\gamma_{7}}^{\gamma_{7}} \\ \le& \frac{s^{2}}{2}-\min\{b_{3}, b_{4}\} \bigl[ \bigl\| u^{1}\bigr\| _{\gamma _{6}}^{\gamma_{6}} + \bigl(\min\bigl\{ \bigl\| u^{2}\bigr\| _{\gamma_{7}}, 1\bigr\} \bigr)^{\gamma_{7}} \bigr] \\ \le& \frac{s^{2}}{2}-2^{1-\gamma_{6}}\min\{b_{3}, b_{4}\} \bigl[\bigl\| u^{1}\bigr\| _{\gamma_{6}} + \bigl(\min\bigl\{ \bigl\| u^{2}\bigr\| _{\gamma_{7}}, 1\bigr\} \bigr) \bigr]^{\gamma_{6}} \\ \le& \frac{s^{2}}{2}-2^{1-\gamma_{6}}\min\{b_{3}, b_{4}\}C_{6}^{-\gamma _{6}}s^{\gamma_{6}} \\ = & \frac{s^{2}}{2}-C_{7}s^{\gamma_{6}}, \quad \forall u=u^{-}+u^{0}+se\in X, s\in(0, 1), \end{aligned}$$
which implies that there exist \(s_{0}\in(0, 1)\) and \(\sigma_{0}>0\) such that (3.23) holds.
Step 4. If \(E^{-}\oplus E^{0}=\{0\}\), then Lemmas 2.9 and 3.2, Steps 1-3 imply that Φ has a minimum (<0) which yields a homoclinic solution for system (1.1).
If \(E^{-}\oplus E^{0}\ne\{0\}\), by Step 2, one can take \(C_{8}>0\) and \(r>s_{0}\) large such that
$$\Phi(u)\ge-C_{8}, \quad \forall u\in E^{+} $$
and
$$\Phi(u)\ge0, \quad \forall u\in E^{+} \mbox{ with } \|u\| \ge r. $$
Let \(Q=B_{r}\cap E^{+}\). Since \(S_{e}\) and ∂Q link, by Lemma 2.7, −Φ has a critical point \(u^{*}\in E\) with \(\Phi(u^{*})\le-\sigma_{0}\), which is a nontrivial homoclinic solution of system (1.1). □
Proof of Theorem 1.2
Set \(X=E\), \(X_{1}=E^{-}\oplus E^{0}\) and \(X_{2}=E^{+}\). In view of Lemma 3.2 and Steps 1 and 2 in the proof of Theorem 1.1, \(X=X_{1}\oplus X_{2}\), \(\dim X_{1}<+\infty\), \(\Phi\in C^{1}(X, \mathbb {R})\), Φ satisfies the (PS)-condition and is bounded from below on \(X_{2}\). Obviously, (W1) and (W5) imply \(\Phi(0)=0\) and Φ is even. Next, we prove that assumption (ii) in Lemma 2.8 holds.
Let \(\tilde{X}\subset X\) be any finite dimensional subspace. Then there exist constants \(c_{0}=c(\tilde{X})>0\) and \(c_{*}=c(\tilde{X})>0\) such that
$$ c_{0}\|u\|\le\|u\|_{\gamma_{6}}, \|u \|_{\gamma_{7}}, \|u\|_{\infty}\le c_{*}\|u\|, \quad \forall u\in\tilde{X}. $$
(3.26)
Since \(\gamma_{6}\ge\gamma_{7}\), it follows from (3.7) and (3.26) that
$$ \begin{aligned}[b] \|u\|_{\gamma_{6}}^{\gamma_{6}} & = \bigl\| u^{1} \bigr\| _{\gamma_{6}}^{\gamma_{6}}+\bigl\| u^{2}\bigr\| _{\gamma_{6}}^{\gamma_{6}} \le\bigl\| u^{1}\bigr\| _{\gamma_{6}}^{\gamma_{6}}+\|u\|_{\infty}^{\gamma_{6}-\gamma _{7}} \bigl\| u^{2}\bigr\| _{\gamma_{7}}^{\gamma_{7}} \\ & \le \bigl\| u^{1}\bigr\| _{\gamma_{6}}^{\gamma_{6}}+\bigl\| u^{2} \bigr\| _{\gamma_{7}}^{\gamma _{7}}, \quad \forall u\in\tilde{X}, c_{*}\|u\|< 1. \end{aligned} $$
(3.27)
From (3.5), (3.7), (3.26), (3.27) and (W4), one has
$$\begin{aligned} \Phi(u) = & \frac{1}{2} \bigl(\bigl\| u^{+}\bigr\| ^{2}- \bigl\| u^{-}\bigr\| ^{2} \bigr)-\int_{\mathbb {R}}W(t, u)\, \mathrm{d}t \\ \le& \frac{1}{2}\|u\|^{2}-b_{3}\bigl\| u^{1} \bigr\| _{\gamma_{6}}^{\gamma_{6}}-b_{4}\bigl\| u^{2} \bigr\| _{\gamma_{7}}^{\gamma_{7}} \\ \le& \frac{1}{2}\|u\|^{2}-\min\{b_{3}, b_{4}\}\|u\|_{\gamma_{6}}^{\gamma _{6}} \\ \le& \frac{1}{2}\|u\|^{2}-c_{0}^{\gamma_{6}} \min\{b_{3}, b_{4}\}\|u\| ^{\gamma_{6}},\quad \forall u \in\tilde{X}, c_{*}\|u\|< 1. \end{aligned}$$
Since \(1< \gamma_{6} <2\), the above implies that there exist \(\rho=\rho (b_{3}, b_{4}, c_{0})=\rho(\tilde{X})\in(0, c_{*}^{-1})\) and \(\sigma=\sigma(b_{3}, b_{4}, c_{0})=\sigma(\tilde{X})>0\) such that
$$\Phi(u)\le0, \quad \forall u\in B_{\rho}\cap\tilde{X} ; \qquad \Phi(u) \le-\sigma, \quad \forall u\in\partial B_{\rho}\cap \tilde{X}. $$
Hence assumption (ii) in Lemma 2.8 holds. By Lemma 2.8, Φ has infinitely many (pairs) critical points which are homoclinic solutions for system (1.1). □
Proof of Theorem 1.3
In the proof of Theorem 1.1, assumption (W3) is used only in Step 1 to prove that a (PS)-sequence \(\{u_{n}\}_{n\in{\mathbb{N}}}\subset E\) is bounded. Therefore, we only prove that any (PS)-sequence \(\{u_{n}\}_{n\in{\mathbb{N}}}\subset E\) is also bounded by using (W3′) instead of (W3). From (3.5), (3.6) and (W3′), we have
$$\begin{aligned} \bigl\langle \Phi'(u_{n}), u_{n} \bigr\rangle -2\Phi(u_{n}) = & \int_{\mathbb {R}}\bigl[2W(t, u_{n})-\bigl(\nabla W(t, u_{n}), u_{n}\bigr) \bigr]\,\mathrm{d}t \\ \ge& b_{5}\int_{\mathbb {R}}\bigl|u_{n}(t)\bigr|^{\gamma_{8}} \,\mathrm{d}t - b_{6}\int_{\mathbb {R}}|u_{n}|^{\gamma_{9}} \,\mathrm{d}t \\ = & b_{5}\|u_{n}\|_{\gamma_{8}}^{\gamma_{8}}-b_{6} \|u_{n}\|_{\gamma_{9}}^{\gamma_{9}}. \end{aligned}$$
It follows that there exists a constant \(C_{9}>0\) such that
$$ b_{5}\|u_{n}\|_{\gamma_{8}}^{\gamma_{8}}-b_{6} \|u_{n}\|_{\gamma_{9}}^{\gamma _{9}}\le C_{9}\bigl(1+ \|u_{n}\|\bigr). $$
(3.28)
Since \(\dim(E^{-}\oplus E^{0})<+\infty\), there exists a constant \(C_{10}>0\) such that
$$ \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}^{2} = \bigl(u_{n}^{-}+u_{n}^{0}, u_{n} \bigr)_{2} \le\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{\gamma_{8}'}\|u_{n}\|_{\gamma_{8}} \le C_{10} \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}\|u_{n}\|_{\gamma_{8}}, $$
(3.29)
where \(\gamma_{8}'=\gamma_{8}/(\gamma_{8}-1)\). Combining (3.28) with (3.29), one has
$$ \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}\le C_{11}\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}^{2}\le C_{12} \bigl(1+\|u_{n} \|^{2/\gamma_{8}} +\|u_{n}\|^{2\gamma_{9}/\gamma_{8}} \bigr). $$
(3.30)
From (3.5), (3.11) and (3.30), we obtain
$$\begin{aligned} \|u_{n}\|^{2} = & \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}+\bigl\| u_{n}^{+}\bigr\| ^{2} \\ = & \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}+2\Phi(u_{n})+\bigl\| u_{n}^{-} \bigr\| ^{2}+2\int_{\mathbb {R}}W(t, u_{n})\, \mathrm{d}t \\ \le& 2C_{12} \bigl(1+\|u_{n}\|^{2/\gamma_{8}}+ \|u_{n}\|^{2\gamma _{9}/\gamma _{8}} \bigr)+2\Phi(u_{n}) \\ &{} +2\phi_{1}(T_{2})\|u_{n}\|^{\gamma_{1}}+2 \phi_{2}(T_{2})\|u_{n}\|^{\gamma _{2}} \\ \le& C_{13} \bigl(1+\|u_{n}\|^{\gamma_{1}}+ \|u_{n}\|^{\gamma_{2}}+\|u_{n}\| ^{2/\gamma_{8}}+ \|u_{n}\|^{2\gamma_{9}/\gamma_{8}} \bigr). \end{aligned}$$
Since \(1<\gamma_{1}<\gamma_{2}<2\), \(1< \gamma_{9} < \gamma_{8}<2\), it follows that \(\{\|u_{n}\|\}\) is bounded. The proof is complete. □
Proof of Theorem 1.4
Set \(X=E\), \(X_{1}=E^{-}\oplus E^{0}\) and \(X_{2}=E^{+}\). In view of Lemma 3.2 and Steps 1 and 2 in the proof of Theorem 1.1, \(X=X_{1}\oplus X_{2}\), \(\dim X_{1}<+\infty\), \(\Phi\in C^{1}(X, \mathbb {R})\), Φ satisfies the (PS)-condition and is bounded from below on \(X_{2}\). Obviously, (W1) and (W5) imply \(\Phi(0)=0\) and Φ is even. Next, we prove that assumption (ii) in Lemma 2.8 holds.
Let \(\tilde{X}\subset X\) be any finite dimensional subspace. Then there exist constants \(c_{0}=c(\tilde{X})>0\) and \(c_{*}=c(\tilde{X})>0\) such that
$$ c_{0}\|u\|\le\|u\|_{\gamma_{10}}, \|u \|_{\gamma_{11}}\le c_{*}\|u\|, \quad \forall u\in\tilde{X}. $$
(3.31)
From (3.5), (3.31) and (W4′), one has
$$\begin{aligned} \Phi(u) = & \frac{1}{2} \bigl(\bigl\| u^{+}\bigr\| ^{2}- \bigl\| u^{-}\bigr\| ^{2} \bigr)-\int_{\mathbb {R}}W(t, u)\, \mathrm{d}t \\ \le& \frac{1}{2}\|u\|^{2}-b_{7}\|u \|_{L^{\gamma_{10}}}^{\gamma_{10}} + b_{8}\|u\|_{L^{\gamma_{11}}}^{\gamma_{11}} \\ \le& \frac{1}{2}\|u\|^{2}-b_{7}c_{0}^{\gamma_{10}} \|u\|^{\gamma_{10}} + b_{8}c_{*}^{\gamma_{11}}\|u\|^{\gamma_{11}}, \quad \forall u\in\tilde{X}. \end{aligned}$$
Since \(1< \gamma_{10}< \gamma_{11} <2\), the above implies that there exist \(\rho=\rho(b_{7}, b_{8}, c_{0})=\rho(\tilde{X})>0\) and \(\sigma=\sigma(b_{7}, b_{8}, c_{0})=\sigma(\tilde{X})>0\) such that
$$\Phi(u)\le0, \quad \forall u\in B_{\rho}\cap\tilde{X} ; \qquad \Phi(u) \le-\sigma, \quad \forall u\in\partial B_{\rho}\cap \tilde{X}. $$
Hence assumption (ii) in Lemma 2.8 holds. By Lemma 2.8, Φ has infinitely many (pairs) critical points which are homoclinic solutions for system (1.1). □
In the proof of Theorem 1.4, (W4′) is used in the last part to verify assumption (ii) of Lemma 2.8. It is easy to see that it also holds by using (W4″) instead of (W4′). So we omit the proof of Corollary 1.5.