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Multiple homoclinic solutions for p-Laplacian Hamiltonian systems with concave–convex nonlinearities

Abstract

The multiplicity of homoclinic solutions is obtained for a class of the p-Laplacian Hamiltonian systems \(\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))-a(t)|u(t)|^{p-2}u(t)+ \nabla W(t,u(t))=0\) via variational methods, where \(a(t)\) is neither coercive nor bounded necessarily and \(W(t,u)\) is under new concave–convex conditions. Recent results in the literature are generalized even for \(p=2\).

Introduction

Let us consider the p-Laplacian Hamiltonian systems

$$ \frac{d}{dt} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p-2}\dot{u}(t) \bigr)-a(t) \bigl\vert u(t) \bigr\vert ^{p-2}u(t)+ \nabla W \bigl(t,u(t) \bigr)=0, $$
(1)

where \(t\in \mathbb{R}\), \(u\in \mathbb{R}^{N}\), \(p>1\), \(a\in C( \mathbb{R}, [a_{0},+\infty ))\) with \(a_{0}>0\) and \(W\in C^{1}( \mathbb{R}\times \mathbb{R}^{N}, \mathbb{R})\). As usual, we say that u is a nontrivial homoclinic solution (to 0) if \(u\not \equiv 0\), \(u(t)\) and \(\dot{u}(t)\to 0\) as \(|t|\to +\infty \).

If \(p\equiv 2\) and \(a(t)=L(t)\), (1) reduces to the second order Hamiltonian system

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

where \(L\in C(\mathbb{R}, \mathbb{R}^{N^{2}})\) is a symmetric and positive definite matrix for all \(t\in \mathbb{R}\). In the last 30 years, the existence and multiplicity of solutions for Hamiltonian systems or other differential systems have been investigated in many papers via variational methods (see [14, 9, 11, 1418, 23]). It is well-known that homoclinic orbits play an important role in analyzing the chaos of dynamical systems. Since the problem is considered on the whole space, one of the difficulties to find the solutions of Hamiltonian systems is the lack of compactness of the Sobolev embedding. To overcome this difficulty, \(L(t)\) and \(W(t,x)\) were assumed to be periodic in t. Without periodicity, Rabinowitz and Tanaka [9] introduced the following coercive condition:

(L):

there exists a continuous function \(\alpha :\mathbb{R} \to \mathbb{R}^{+}\) satisfying

$$\begin{aligned} \bigl(L(t)x,x \bigr)\geq \alpha (t) \vert x \vert ^{2}\quad \text{and}\quad \alpha (t)\to +\infty \quad \text{as } \vert t \vert \to +\infty . \end{aligned}$$

The operator \(\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))\) in (1) is said to be p-Laplacian. In the last decade there has been an increasing interest in the study of ordinary differential systems driven by the p-Laplacian. The existence and multiplicity of homoclinic orbits for the p-Laplacian Hamiltonian system were studied in recent papers [57, 10, 12, 13, 19, 20, 22] and the references therein. Similarly, to overcome the lack of compactness of the Sobolev embedding, the following coercive assumption on a was assumed in [5]:

(A):

a is a positive continuous function such that

$$\begin{aligned} a(t)\to +\infty \quad \text{as } \vert t \vert \to +\infty . \end{aligned}$$
It is clear that the coercive conditions are much restrictive. In a recent paper, Zhang et al. [22] proved the existence of two nontrivial homoclinic solutions of problem (1) without coercive conditions. They assumed that a is bounded, that is,
(\(A'\)):

there are two constants \(\tau _{1}\) and \(\tau _{2}\) such that

$$\begin{aligned} 0< \tau _{1}\leq a(t)\leq \tau _{2}< +\infty \quad \text{for all } t\in \mathbb{R}. \end{aligned}$$
Besides, they considered the concave–convex nonlinearity, which is of the form

$$\begin{aligned} W(t,x)=W_{1}(t,x)+W_{2}(t,x), \end{aligned}$$

where \(W_{1}\) is of super-p growth at infinity and \(W_{2}\) is of sub-p growth at infinity. Explicitly, the authors supposed the following conditions:

(\(V_{1}\)):

there exists a constant \(\vartheta >{p}\) such that

$$\begin{aligned} 0< \vartheta W_{1}(t,x)\leq \bigl(\nabla W_{1}(t,x),x \bigr),\quad \forall (t,x) \in \mathbb{R}\times \mathbb{R}^{N} \setminus \{0\}; \end{aligned}$$
(\(V_{2}\)):

there exists a continuous function \(w:\mathbb{R} \to \mathbb{R}^{+}\) such that

$$\begin{aligned} \lim_{|t|\to +\infty } w(t)=0 \end{aligned}$$

and

$$\begin{aligned} \bigl\vert \nabla W_{1}(t,x) \bigr\vert \leq w(t) \vert x \vert ^{\vartheta -1}\quad \text{for all } (t,x) \in \mathbb{R}\times \mathbb{R}^{N}; \end{aligned}$$
(\(V_{3}\)):

\(W_{2}(t,0)=0\) for all \(t\in \mathbb{R}\), \(W_{2} \in C^{1}(\mathbb{R}\times \mathbb{R}^{N}, \mathbb{R})\) and there exist a constant \(1<\varrho <2\) and a continuous function \(b:\mathbb{R} \to \mathbb{R}^{+}\) such that

$$\begin{aligned} W_{2}(t,x)\geq b(t) \vert x \vert ^{\varrho } \end{aligned}$$

for all \((t,x)\in \mathbb{R}\times \mathbb{R}^{N}\);

(\(V_{4}\)):

for all \(t\in \mathbb{R}\) and \(x\in \mathbb{R}^{N}\),

$$\begin{aligned} \bigl\vert \nabla W_{2}(t,x) \bigr\vert \leq c(t) \vert u \vert ^{\varrho -1}, \end{aligned}$$

where \(c: \mathbb{R}\to \mathbb{R}^{+}\) is a continuous function such that \(c\in L^{\xi }(\mathbb{R},\mathbb{R})\) for some constant \(1\leq \xi \leq 2\);

(\(V_{5}\)):
$$\begin{aligned} \biggl(\frac{p \Vert c \Vert _{\xi }C^{\varrho }_{\varrho \xi ^{*}}}{\varrho }\frac{ \vartheta -\varrho }{\vartheta -p} \biggr)^{\vartheta -p}< \biggl(\frac{ \vartheta }{p \Vert \omega \Vert _{\infty }C^{\vartheta }_{\vartheta }}\frac{p- \varrho }{\vartheta -\varrho } \biggr)^{\varrho -p}, \end{aligned}$$

where \(\xi ^{*}\) is the conjugate component of ξ.

Obviously, we can deduce from the conditions \((V_{1})\) and \((V_{2})\) that
(\(W_{0}\)):

there exist constants \(c_{1}, c_{2}>0\) and \(\mu >p\) such that

$$\begin{aligned} \bigl\vert \nabla W_{1}(t,x) \bigr\vert \leq c_{1} \vert x \vert ^{\mu -1}+c_{2}\quad \text{for all } (t,x) \in \mathbb{R}\times \mathbb{R}^{N}; \end{aligned}$$
(\(W_{1}\)):

\(\nabla W_{1}(t,x)=o(|x|^{{p}-1})\) as \(|x|\to 0\) uniformly in t;

(\(W_{2}\)):

\(W_{1}(t,x)/|x|^{{p}}\to +\infty \) as \(|x|\to + \infty \) uniformly in t;

(\(W_{3}\)):

there exists \(d_{1}>0\) such that \(W_{1}(t,x)\geq -d _{1}|x|^{{p}}\) for all \((t,x)\in \mathbb{R}\times \mathbb{R}^{N}\);

(\(W_{4}\)):

there are constants \(\nu >p\) and \(\rho _{0}\), \(d_{2}>0\) such that

$$\begin{aligned} \bigl(\nabla W_{1}(t,x),x \bigr)-\nu W_{1}(t,x)\geq -d_{2} \vert x \vert ^{p},\quad \forall t \in \mathbb{R}, \forall \vert x \vert \geq \rho _{0}. \end{aligned}$$

Motivated by the above facts, in this note, we try to drop both conditions \((A)\) and \((A')\) and consider the following conditions:

(\(A_{1}\)):

\(\int _{\mathbb{R}} a(t)^{-\frac{q}{p}}\,dt<+\infty \), where q is the conjugate component of p, that is, \(\frac{1}{p}+ \frac{1}{q}=1\);

(\(A_{2}\)):

there exists a constant \(\lambda >q^{-1}\) such that

$$\begin{aligned} \operatorname{meas} \bigl(t\in \mathbb{R}|\ \vert t \vert ^{-\lambda p}a(t)< M \bigr)< + \infty , \quad \forall M>0, \end{aligned}$$

where \(\text{meas}(\cdot )\) denotes the Lebesgue measure and q is the conjugate component of p.

Using conditions (\(A_{1}\)) and (\(A_{2}\)) separately, we prove some new compact embedding theorems and discuss the multiplicity of homoclinic solutions for problem (1) with weaker combined nonlinearities. Now we state our main results.

Theorem 1

Suppose that\(W(t,x)=W_{1}(t,x)+W_{2}(t,x)\). Assume\((A_{1})\), (\(W _{0}\))(\(W_{4}\)) and the following conditions hold:

(\(W_{5}\)):

\(W_{2}(t,0)=0\)for all\(t\in \mathbb{R}\)and there exist a constant\(1<\theta <p\)and a continuous function\(b: \mathbb{R}\to \mathbb{R}^{+}\)such that

$$\begin{aligned} W_{2}(t,x)\geq b(t) \vert x \vert ^{\theta } \end{aligned}$$

for all\((t,x)\in \mathbb{R}\times \mathbb{R}^{N}\);

(\(W_{6}\)):

\(W_{2}\in C^{1}(\mathbb{R}\times \mathbb{R}^{N}, \mathbb{R})\)and there exists a continuous function\(c: \mathbb{R} \to \mathbb{R}^{+}\)such that

$$\begin{aligned} \bigl\vert \nabla W_{2}(t,x) \bigr\vert \leq c(t) \vert x \vert ^{\theta -1}, \end{aligned}$$

where\(c\in L^{\zeta }( \mathbb{R}, \mathbb{R})\)for some constant\(\zeta >1\)and\(\|c\|_{\zeta }\)is small enough;

(\(W_{7}\)):

\(\zeta ^{*}(\theta -1)\geq p\), where\(\zeta ^{*}\)is the conjugate component ofζ.

Then problem (1) possesses at least two nontrivial homoclinic solutions.

Remark 1

From Theorem 1, we see that the conditions related to the sup-p term \(W_{1}\) are weaker than that in [22]. There are functions satisfying the conditions (\(W_{0}\))–(\(W_{4}\)) but not (\(V_{1}\)) and (\(V_{2}\)). Moreover, we can also give some examples of a not satisfying the conditions (A) and (\(A'\)). For example, let

$$ W_{1}(t,x)= \textstyle\begin{cases} - \vert x \vert ^{4}+ \vert x \vert ^{3}, & \vert x \vert \leq \frac{4}{5} \\ ( \vert x \vert -\frac{4+4^{\frac{1}{3}}}{5} )^{4}+\frac{64-4^{ \frac{4}{3}}}{625}, & \vert x \vert \geq \frac{4}{5}, \end{cases}\displaystyle \qquad W_{2}(t,x)= \frac{\epsilon }{ (1+t^{2} )^{\frac{3}{4}}}\frac{2 \vert x \vert ^{ \frac{3}{2}}}{3}, $$

and

$$\begin{aligned} a(t)= \textstyle\begin{cases} (n^{2}+1)^{2}( \vert t \vert -n)+c_{0}, & n\leq \vert t \vert < n+\frac{1}{n^{2}+1}, \\ (n^{2}+1)+c_{0}, & n+\frac{1}{n^{2}+1}\leq \vert t \vert < n+\frac{n^{2}}{n^{2}+1}, \\ (n^{2}+1)^{2}(n+1- \vert t \vert )+c_{0}, & n+\frac{n^{2}}{n^{2}+1}\leq \vert t \vert < n+1, \end{cases}\displaystyle \end{aligned}$$

where \(n\in \mathbb{N}\), \(c_{0}\in \mathbb{R}\). A straightforward computation shows that \(W_{1}\), \(W_{2}\) and a satisfy the assumptions of Theorem 1 with \(p=2\), \(\mu =5\), \(\theta =\frac{3}{2}\), \(\zeta = \frac{4}{3}\) and \(\epsilon >0\) small enough.

By replacing the condition \((A_{1})\), we have the following theorem.

Theorem 2

Assume that\(W(t,x)=W_{1}(t,x)+W_{2}(t,x)\). Suppose that\((A_{2})\)and (\(W_{0}\))(\(W_{7}\)) hold, then problem (1) possesses at least two nontrivial homoclinic solutions.

Remark 2

There exist functions that satisfy the condition (\(A_{2}\)) but do not satisfy the conditions (A) and (\(A'\)), such as \(a(t)=t^{4}{\sin } ^{2}t+1\) with \(p=2\) and \(\lambda =1\). Thus Theorem 2 is different from the previous results.

Proof of Theorem 1

First, we introduce the space in which we can construct the variational framework. Let

$$\begin{aligned} E= \biggl\{ u\in W^{1,p} \bigl(\mathbb{R}, \mathbb{R}^{N} \bigr): \int _{\mathbb{R}} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p}+a(t) \bigl\vert u(t) \bigr\vert ^{p} \bigr) \,dt< +\infty \biggr\} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u \Vert = \biggl( \int _{\mathbb{R}} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p}+a(t) \bigl\vert u(t) \bigr\vert ^{p} \bigr) \,dt \biggr) ^{\frac{1}{p}}. \end{aligned}$$

Then E is a uniform convex Banach space. Denote by \(L^{\gamma }( \mathbb{R}, \mathbb{R}^{N})\) (\(1\leq \gamma <+\infty \)) the Banach spaces of functions with the norms

$$\begin{aligned} \Vert u \Vert _{\gamma }= \biggl( \int _{\mathbb{R}} \bigl\vert u(t) \bigr\vert ^{\gamma } \,dt \biggr) ^{\frac{1}{\gamma }}, \end{aligned}$$

and \(L^{\infty }(\mathbb{R}, \mathbb{R}^{N})\) is the Banach space of essentially bounded functions under the norm

$$\begin{aligned} \Vert u \Vert _{\infty }=\operatorname{ess}\ \sup \bigl\{ \bigl\vert u(t) \bigr\vert : t\in \mathbb{R} \bigr\} . \end{aligned}$$

Lemma 1

([22])

The embedding\(E\hookrightarrow L^{\gamma }( \mathbb{R},\mathbb{R}^{N})\) (\(p\leq \gamma \leq +\infty \)) is continuous.

Lemma 2

Under the condition\((A_{1})\), the embedding\(E\hookrightarrow L^{1}( \mathbb{R},\mathbb{R}^{N})\)is continuous and compact.

Proof

By \((A_{1})\) and Hölder’s inequality, for all \(u\in E\) one has

$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert u(t) \bigr\vert \,dt =& \int _{\mathbb{R}}a(t)^{-\frac{1}{p}}a(t)^{ \frac{1}{p}} \bigl\vert u(t) \bigr\vert \,dt \\ \leq & \biggl( \int _{\mathbb{R}} a(t)^{-\frac{q}{p}}\,dt \biggr)^{ \frac{1}{q}} \biggl( \int _{\mathbb{R}} a(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt \biggr)^{ \frac{1}{p}} \\ \leq & \biggl( \int _{\mathbb{R}} a(t)^{-\frac{q}{p}}\,dt \biggr)^{ \frac{1}{q}} \Vert u \Vert , \end{aligned}$$

which implies that the embedding is continuous.

Let \(\{u_{n}\}\subset E\) be a sequence such that \(u_{n}\rightharpoonup 0\) in E. By Banach–Steinhaus Theorem, there exists \(M_{0}>0\) such that

$$\begin{aligned} \sup_{n\in \mathbb{N}} \Vert u_{n} \Vert \leq M_{0}. \end{aligned}$$

Since the embedding is compact on bounded domain, it suffices to show that, for any \(\varepsilon >0\), there exists \(r>0\) such that

$$\begin{aligned} \int _{ \vert t \vert > r} \bigl\vert u_{n}(t) \bigr\vert \,dt< \varepsilon . \end{aligned}$$

In fact, we have

$$\begin{aligned} \int _{ \vert t \vert > r} \bigl\vert u_{n}(t) \bigr\vert \,dt \leq & \int _{ \vert t \vert >r}a(t)^{-\frac{1}{p}}a(t)^{ \frac{1}{p}} \bigl\vert u_{n}(t) \bigr\vert \,dt \\ \leq & \biggl( \int _{ \vert t \vert >r} a(t)^{-\frac{q}{p}}\,dt \biggr)^{\frac{1}{q}} \biggl( \int _{ \vert t \vert >r}a(t) \bigl\vert u_{n}(t) \bigr\vert ^{p} \,dt \biggr)^{\frac{1}{p}} \\ \leq & \biggl( \int _{ \vert t \vert >r} a(t)^{-\frac{q}{p}}\,dt \biggr)^{\frac{1}{q}} \Vert u_{n} \Vert \\ \leq & \biggl( \int _{ \vert t \vert >r} a(t)^{-\frac{q}{p}}\,dt \biggr)^{\frac{1}{q}}M _{0}. \end{aligned}$$

It follows from (\(A_{1}\)) that this can be made arbitrarily small by choosing r large. Hence, we get \(u_{n}\to 0\) in \(L^{1}(\mathbb{R}, \mathbb{R}^{N})\). □

Remark 3

From Lemma 1 and Lemma 2, for \(\gamma =1\) or \(p\leq \gamma \leq + \infty \), there exists \(C_{\gamma }>0\) such that

$$\begin{aligned} \Vert u \Vert _{\gamma }\leq C_{\gamma } \Vert u \Vert , \quad \forall u\in E. \end{aligned}$$
(2)

Lemma 3

Suppose that the conditions\((A_{1})\)and\((W_{1})\)hold, then we have\(\nabla W_{1}(t,u_{n})\to \nabla W_{1}(t,u)\)in\(L^{{q}}(\mathbb{R}, \mathbb{R}^{N})\)if\(u_{n}\rightharpoonup u\)inE.

Proof

Assume that \(u_{n}\rightharpoonup u\) in E. By the Banach–Steinhaus theorem and (2), there exists \(M_{1}>0\) such that

$$\begin{aligned} \sup_{n\in \mathbb{N}} \Vert u_{n} \Vert _{\infty }\leq M_{1} \quad \text{and}\quad \Vert u \Vert _{\infty }\leq M_{1}. \end{aligned}$$
(3)

We can deduce from \((W_{0})\), \((W_{1})\) and (3) that there exists \(M_{2}>0\) such that

$$\begin{aligned} \bigl\vert \nabla W_{1}(t,u_{n}) \bigr\vert \leq M_{2} \bigl\vert u_{n}(t) \bigr\vert ^{{p}-1} \quad \text{and}\quad \bigl\vert \nabla W_{1}(t,u) \bigr\vert \leq M_{2} \bigl\vert u(t) \bigr\vert ^{{p}-1}, \end{aligned}$$

which implies that

$$\begin{aligned} \bigl\vert \nabla W_{1}(t,u_{n})-\nabla W_{1}(t,u) \bigr\vert \leq & M_{2} \bigl( \bigl\vert u_{n}(t) \bigr\vert ^{ {p}-1}+ \bigl\vert u(t) \bigr\vert ^{{p}-1} \bigr) \\ \leq &M_{2} \bigl[2^{{p}-1} \bigl( \bigl\vert u_{n}(t)-u(t) \bigr\vert ^{{p}-1}+ \bigl\vert u(t) \bigr\vert ^{{p}-1} \bigr)+ \bigl\vert u(t) \bigr\vert ^{ {p}-1} \bigr] \\ \leq &M_{3} \bigl( \bigl\vert u_{n}(t)-u(t) \bigr\vert ^{{p}-1}+ \bigl\vert u(t) \bigr\vert ^{{p}-1} \bigr), \end{aligned}$$
(4)

where \(M_{3}\) is a positive constant. By (2), (3), (4) and Lemma 2 one gets

$$\begin{aligned}& \int _{\mathbb{R}} \bigl\vert \nabla W_{1}(t,u_{n})- \nabla W_{1}(t,u) \bigr\vert ^{{q}}\,dt \\& \quad \leq M_{3}^{q} \int _{\mathbb{R}} \bigl( \bigl\vert u_{n}(t)-u(t) \bigr\vert ^{{p}-1}+ \bigl\vert u(t) \bigr\vert ^{ {p}-1} \bigr)^{{q}}\,dt \\& \quad \leq 2^{{q}-1}M_{3}^{q} \int _{\mathbb{R}} \bigl( \bigl\vert u_{n}(t)-u(t) \bigr\vert ^{{p}}+ \bigl\vert u(t) \bigr\vert ^{p} \bigr)\,dt \\& \quad \leq 2^{{q}-1}M_{3}^{{q}} \Vert u_{n}-u \Vert ^{p-1}_{\infty } \int _{ \mathbb{R}} \bigl\vert u_{n}(t)-u(t) \bigr\vert \,dt+2^{{q}-1}M_{3}^{{q}} \Vert u \Vert ^{p}_{p} \\& \quad \leq 2^{{q}-1}M_{3}^{{q}}(2M_{1})^{p-1} \int _{\mathbb{R}} \bigl\vert u_{n}(t)-u(t) \bigr\vert \,dt+2^{ {q}-1}M_{3}^{{q}}C^{p}_{p} \Vert u \Vert ^{p} \\& \quad < +\infty . \end{aligned}$$

Using Lebesgue’s dominated convergence theorem, we can get the conclusion. □

The corresponding functional of (1) is defined by

$$\begin{aligned} I(u) =& \int _{\mathbb{R}}\frac{1}{p} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p}+a(t) \bigl\vert u(t) \bigr\vert ^{p} \bigr)\,dt- \int _{\mathbb{R}} W \bigl(t,u(t) \bigr)\,dt \\ =&\frac{1}{p} \Vert u \Vert ^{p}- \int _{\mathbb{R}} W \bigl(t,u(t) \bigr)\,dt. \end{aligned}$$
(5)

For convenience, let

$$\begin{aligned} J(u) =& \int _{\mathbb{R}}\frac{1}{p} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p}+a(t) \bigl\vert u(t) \bigr\vert ^{p} \bigr)\,dt, \\ \varPhi (u) =& \int _{\mathbb{R}} W_{1} \bigl(t,u(t) \bigr)\,dt, \\ \varPsi (u) =& \int _{ \mathbb{R}} W_{2} \bigl(t,u(t) \bigr)\,dt. \end{aligned}$$

Lemma 4

  1. (i)

    \(J\in C^{1}(E,\mathbb{R})\) and

    $$\begin{aligned} \bigl\langle J'(u),v \bigr\rangle = \int _{\mathbb{R}} \bigl[ \bigl\vert \dot{u}(t) \bigr\vert ^{p-2} \bigl( \dot{u}(t),\dot{v}(t) \bigr)+a(t) \bigl\vert u(t) \bigr\vert ^{p-2} \bigl(u(t),v(t) \bigr) \bigr] \,dt,\quad \forall u,v\in E. \end{aligned}$$
  2. (ii)

    Under the conditions of Theorem 1, \(I\in C^{1}(E, \mathbb{R})\). Moreover, one has

    $$\begin{aligned} \bigl\langle I'(u),v \bigr\rangle =& \int _{\mathbb{R}} \bigl[ \bigl\vert \dot{u}(t) \bigr\vert ^{p-2} \bigl( \dot{u}(t),\dot{v}(t) \bigr)+a(t) \bigl\vert u(t) \bigr\vert ^{p-2} \bigl(u(t),v(t) \bigr) \\ &{}- \bigl(\nabla W \bigl(t,u(t) \bigr),v(t) \bigr) \bigr]\,dt, \quad \forall u,v \in E. \end{aligned}$$
    (6)
  3. (iii)

    The critical points ofIinEare homoclinic solutions of (1) with\(u(\pm \infty )=\dot{u}(\pm \infty )=0\).

Proof

Since it is routine to prove that (i) holds, we just need to prove (ii) and (iii). First, we show I in (5) is well defined. By \((W_{0})\) and \((W_{1})\), for any \(\varepsilon >0\), there is \(C_{\varepsilon }>0\) such that

$$\begin{aligned} \bigl\vert W_{1}(t,x) \bigr\vert \leq \varepsilon \vert x \vert ^{p}+C_{\varepsilon } \vert x \vert ^{\mu }, \quad \forall (t,x)\in \mathbb{R}\times \mathbb{R}^{N}. \end{aligned}$$
(7)

Then by (2) and (7) one gets

$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert W_{1} \bigl(t,u(t) \bigr) \bigr\vert \,dt \leq & \varepsilon \int _{ \mathbb{R}} \bigl\vert u(t) \bigr\vert ^{p} \,dt+C_{\varepsilon } \int _{\mathbb{R}} \bigl\vert u(t) \bigr\vert ^{ \mu } \,dt \leq \varepsilon C^{p}_{p} \Vert u \Vert ^{p} +C_{\varepsilon }C^{\mu }_{ \mu } \Vert u \Vert ^{\mu }< +\infty . \end{aligned}$$

Besides, by (2), \((W_{6})\), \((W_{7})\) and Hölder’s inequality we have

$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert W_{2} \bigl(t,u(t) \bigr) \bigr\vert \,dt \leq &\frac{1}{\theta } \int _{\mathbb{R}} c(t) \bigl\vert u(t) \bigr\vert ^{\theta }\,dt \\ \leq &\frac{1}{\theta } \Vert c \Vert _{\zeta } \Vert u \Vert ^{\theta }_{\theta \zeta ^{*}} \\ \leq &\frac{C^{\theta }_{\theta \zeta ^{*}}}{\theta } \Vert c \Vert _{\zeta } \Vert u \Vert ^{\theta }< +\infty . \end{aligned}$$
(8)

Therefore I is well defined. Next, we show that \(I\in C^{1}(E, \mathbb{R})\). In view of (i), it is sufficient to show that \(\varPhi \in C^{1}(E,\mathbb{R})\) and \(\varPsi \in C^{1}(E,\mathbb{R})\). Let \(\phi (u)\) be as follows:

$$\begin{aligned} \phi (u)v= \int _{\mathbb{R}} \bigl(\nabla W_{1} \bigl(t,u(t) \bigr),v(t) \bigr)\,dt,\quad \forall v \in E. \end{aligned}$$
(9)

Obviously, \(\phi (u)\) is linear. We show \(\phi (u)\) is bounded in the following proof. By (2), (9), \((W_{0})\) and Hölder’s inequality, one has

$$\begin{aligned} \bigl\vert \phi (u)v \bigr\vert \leq & c_{1} \int _{\mathbb{R}} \bigl\vert u(t) \bigr\vert ^{\mu -1} \bigl\vert v(t) \bigr\vert \,dt+c _{2} \int _{\mathbb{R}} \bigl\vert v(t) \bigr\vert \,dt \\ \leq & c_{1} \biggl( \int _{\mathbb{R}} \bigl\vert u(t) \bigr\vert ^{(\mu -1)\mu ^{*}} \,dt \biggr) ^{\frac{1}{\mu ^{*}}} \biggl( \int \bigl\vert v(t) \bigr\vert ^{\mu }\,dt \biggr)^{\frac{1}{ \mu }}+c_{2} \Vert v \Vert _{1} \\ \leq &c_{1} \Vert u \Vert _{\mu }^{\frac{\mu }{\mu ^{*}}} \Vert v \Vert _{\mu }+c_{2}C _{1} \Vert v \Vert \\ \leq & \bigl(c_{1}C_{\mu }^{\frac{\mu }{\mu ^{*}}+1} \Vert u \Vert ^{\frac{ \mu }{\mu ^{*}}}+c_{2}C_{1} \bigr) \Vert v \Vert , \end{aligned}$$
(10)

where \(\mu ^{*}\) is the conjugate component of μ. It follows from (10) that \(\phi (u)\) is bounded. Subsequently, we show that Φ is of \(C^{1}\) class. For any \(u,v\in E\), by the mean value theorem, (\(W_{0}\)) and Hölder’s inequality, one gets

$$\begin{aligned}& \biggl\vert \int _{\mathbb{R}} W_{1} \bigl(t, u(t)+v(t) \bigr)\,dt- \int _{\mathbb{R}}(W _{1} \bigl(t,u(t) \bigr)\,dt \biggr\vert \\& \quad = \biggl\vert \int _{\mathbb{R}}(\nabla W_{1} \bigl(t, u(t)+h(t)v(t),v(t) \bigr)\,dt \biggr\vert \\& \quad \leq c_{1} \int _{\mathbb{R}} \bigl\vert u(t)+h(t)v(t) \bigr\vert ^{\mu -1} \bigl\vert v(t) \bigr\vert \,dt +c_{2} \int _{\mathbb{R}} \bigl\vert v(t) \bigr\vert \,dt \\& \quad \leq c_{1} \Vert u+hv \Vert _{\mu }^{\frac{\mu }{\mu ^{*}}} \Vert v \Vert _{\mu } +c_{2}C_{1} \Vert v \Vert \\& \quad \leq \bigl(c_{1}C_{\mu }^{\frac{\mu }{\mu ^{*}}+1} \Vert u+hv \Vert ^{\frac{ \mu }{\mu ^{*}}}+c_{2}C_{1} \bigr) \Vert v \Vert , \end{aligned}$$
(11)

where \(h(t)\in (0,1)\). Combining (10) and (11), we get

$$\begin{aligned} \int _{\mathbb{R}} W_{1} \bigl(t,u(t)+v(t) \bigr)\,dt- \int _{\mathbb{R}} W_{1} \bigl(t,u(t) \bigr)\,dt- \int _{\mathbb{R}} \bigl(\nabla W_{1} \bigl(t, u(t) \bigr),v(t) \bigr)\,dt\to 0 \end{aligned}$$

as \(v\to 0\) in E, which shows

$$\begin{aligned} \bigl\langle \varPhi '(u),v \bigr\rangle = \int _{\mathbb{R}} \bigl(\nabla W_{1} \bigl(u(t) \bigr),v(t) \bigr)\,dt \end{aligned}$$

for any \(u, v\in E\). It remains to prove that \(\varPhi '\) is continuous. Assume that \(u\to u_{0}\) in E and note that

$$\begin{aligned}& \sup_{ \Vert v \Vert =1} \bigl\vert \bigl\langle \varPhi '(u),v \bigr\rangle - \bigl\langle \varPhi '(u_{0}),v \bigr\rangle \bigr\vert \\& \quad =\sup_{ \Vert v \Vert =1} \biggl\vert \int _{\mathbb{R}} \bigl(\nabla W_{1} \bigl(t, u(t) \bigr)- \nabla W_{1} \bigl(t,u_{0}(t) \bigr),v(t) \bigr) \,dt \biggr\vert \\& \quad \leq \sup_{ \Vert v \Vert =1} \bigl\Vert \nabla W_{1}(t,u)- \nabla W_{1}(t,u_{0}) \bigr\Vert _{q} \biggl( \int _{\mathbb{R}} \bigl\vert v(t) \bigr\vert ^{{p}} \,dt \biggr)^{\frac{1}{p}} \\& \quad \leq \sup_{ \Vert v \Vert =1} \bigl\Vert \nabla W_{1}(t,u)- \nabla W_{1}(t,u_{0}) \bigr\Vert _{q} \biggl( \int _{\mathbb{R}} \bigl\vert v(t) \bigr\vert ^{{p}} \,dt \biggr)^{\frac{1}{p}} \\& \quad \leq C_{p}\sup_{ \Vert v \Vert =1} \bigl\Vert \nabla W_{1}(t,u)-\nabla W_{1}(t,u_{0}) \bigr\Vert _{q}. \end{aligned}$$

Then, by Lemma 3, we have \(\langle \varPhi '(u),v\rangle \to \langle \varPhi '(u_{0}),v\rangle \) as \(\|u\|\to \|u_{0}\|\) uniformly with respect to v, which shows that \(\varPhi '\) is continuous. Moreover, by \((W_{6})\) and \((W_{7})\) one has

$$\begin{aligned} \biggl\vert \int _{\mathbb{R}} \bigl(\nabla W_{2} \bigl(t,u(t) \bigr),v(t) \bigr)\,dt \biggr\vert \leq & \int _{\mathbb{R}} c(t) \bigl\vert u(t) \bigr\vert ^{\theta -1} \bigl\vert v(t) \bigr\vert \,dt \\ \leq & \Vert u \Vert _{\zeta ^{*}(\theta -1)}^{\theta -1} \biggl( \int _{ \mathbb{R}}c^{\zeta }(t)\,dt \biggr)^{\frac{1}{\zeta }} \Vert v \Vert _{\infty } \end{aligned}$$

for any \(u, v\in E\). Similar to the above proof, we can see that

$$\begin{aligned} \bigl\langle \varPsi '(u),v \bigr\rangle = \int _{\mathbb{R}} \bigl(\nabla W_{2} \bigl(u(t) \bigr),v(t) \bigr)\,dt \end{aligned}$$

for any \(u, v\in E\). Now we prove that \(\varPsi '\) is continuous. Suppose that \(u\to u_{0}\) in E. By \((W_{6})\), for any \(\varepsilon >0\), there exists \(T>0\) such that

$$\begin{aligned} \biggl( \int _{|t|>T}c^{\zeta }(t)\,dt \biggr)^{\frac{1}{\zeta }}< \varepsilon . \end{aligned}$$
(12)

On account of the continuity of \(\nabla W_{2}(t,x)\) and \(u\to u_{0}\) in \(L^{\infty }_{\mathrm{loc}}(\mathbb{R},\mathbb{R}^{N})\), it follows that

$$\begin{aligned} \int _{|t|\leq T} \bigl(\nabla W_{2} \bigl(t, u(t) \bigr)- \nabla W_{2} \bigl(t,u_{0}(t) \bigr),v(t) \bigr) \,dt< \varepsilon . \end{aligned}$$
(13)

By (12), (13), \((W_{6})\), \((W_{7})\) and Hölder’s inequality, one gets

$$\begin{aligned}& \sup_{ \Vert v \Vert =1} \bigl\vert \bigl\langle \varPsi '(u),v \bigr\rangle - \bigl\langle \varPsi '(u_{0}),v \bigr\rangle \bigr\vert \\& \quad =\sup_{ \Vert v \Vert =1} \biggl\vert \int _{\mathbb{R}} \bigl(\nabla W_{2} \bigl(t, u(t) \bigr)- \nabla W_{2} \bigl(t,u_{0}(t) \bigr),v(t) \bigr) \,dt \biggr\vert \\& \quad \leq \sup_{ \Vert v \Vert =1} \biggl\vert \int _{ \vert t \vert \leq T} \bigl(\nabla W_{2} \bigl(t, u(t) \bigr)- \nabla W_{2} \bigl(t,u_{0}(t) \bigr),v(t) \bigr) \,dt \biggr\vert \\& \qquad {}+\sup_{ \Vert v \Vert =1} \biggl\vert \int _{ \vert t \vert > T} \bigl(\nabla W_{2} \bigl(t, u(t) \bigr)- \nabla W _{2} \bigl(t,u_{0}(t) \bigr),v(t) \bigr) \,dt \biggr\vert \\& \quad \leq \varepsilon +\sup_{ \Vert v \Vert =1} \biggl\vert \int _{ \vert t \vert >T} c(t) \bigl( \bigl\vert u(t) \bigr\vert ^{ \theta -1}+ \bigl\vert u_{0}(t) \bigr\vert ^{\theta -1} \bigr) \bigl\vert v(t) \bigr\vert \,dt \biggr\vert \\& \quad \leq \varepsilon +C_{\infty } \biggl( \int _{ \vert t \vert >T} c^{\zeta }(t) \,dt \biggr) ^{\frac{1}{\zeta }} \bigl( \Vert u \Vert ^{\theta -1}_{(\theta -1)\zeta ^{*}}+ \Vert u_{0} \Vert ^{\theta -1}_{(\theta -1)\zeta ^{*}} \bigr) \\& \quad \leq \varepsilon +\varepsilon C_{\infty } \bigl( \Vert u \Vert ^{\theta -1}_{( \theta -1)\zeta ^{*}}+ \Vert u_{0} \Vert ^{\theta -1}_{(\theta -1)\zeta ^{*}} \bigr), \end{aligned}$$

which shows that \(\varPsi '\) is continuous. Thus (ii) holds.

Finally, similar to the proof of Lemma 3.1 in [21], one can check that (iii) holds. □

Subsequently, we display the useful critical points theorem.

Lemma 5

([8])

LetEa real Banach space and\(I:E\to \mathbb{R}\)be a\(C^{1}\)-smooth functional and satisfy the\((C)\)condition, that is, \(\{u_{n}\}\)has a convergent subsequence inEwhenever\(\{I(u_{n})\}\)is bounded and\(\|I'(u_{n})\|_{E^{*}}(1+\|u _{n}\|)\to 0\)as\(n\to +\infty \). IfIsatisfies the following conditions:

  1. (i)

    \(I(0)=0\);

  2. (ii)

    there exist constants\(\varrho , \alpha >0\)such that\(I|_{\partial B_{\varrho }(0)}\geq \alpha \);

  3. (iii)

    there exists\(e\in E\setminus \bar{B}_{\varrho }(0)\)such that\(I(e)\leq 0\),

where\(B_{\varrho }(0)\)is an open ball inEof radiusϱcentered at 0, thenIpossesses a critical value\(c\geq \alpha \)given by

$$\begin{aligned} c=\inf_{g\in \varGamma }\max_{s\in [0,1]}I \bigl(g(s) \bigr), \end{aligned}$$

where

$$ \varGamma = \bigl\{ g\in C \bigl([0,1],E \bigr): g(0)=0, g(1)=e \bigr\} . $$

Lemma 6

Assume that the conditions of Theorem 1hold, thenIsatisfies the\((C)\)condition.

Proof

Suppose that \(\{u_{n}\}\subset E\) is a sequence such that \(\{I(u_{n}) \}\) is bounded and \(\|I'(u_{n})\|_{E^{*}}(1+\|u_{n}\|)\to 0\) as \(n\to +\infty \). Then there exists a constant \(M_{4}>0\) such that

$$\begin{aligned} \bigl\vert I(u_{n}) \bigr\vert \leq M_{4}, \qquad \bigl\Vert I'(u_{n}) \bigr\Vert _{E^{*}} \bigl(1+ \Vert u_{n} \Vert \bigr)\leq M_{4}. \end{aligned}$$
(14)

Now we prove that \(\{u_{n}\}\) is bounded in E. Arguing in an indirect way, we assume that \(\|u_{n}\|\to +\infty \) as \(n\to +\infty \). Set \(z_{n}=\frac{u_{n}}{\|u_{n}\|}\), then \(\|z_{n}\|=1\), 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 (2), (5), (8) and (14), we obtain

$$\begin{aligned} \biggl\vert \int _{\mathbb{R}} \frac{W_{1}(t,u_{n})}{ \Vert u_{n} \Vert ^{p}}\,dt- \frac{1}{p} \biggr\vert =& \biggl\vert \frac{I(u_{n})}{ \Vert u_{n} \Vert ^{p}}+ \int _{\mathbb{R}}\frac{W_{2}(t,u_{n})}{ \Vert u_{n} \Vert ^{p}}\,dt \biggr\vert \\ \leq &\frac{M_{4}}{ \Vert u_{n} \Vert ^{p}}+\frac{ \Vert c \Vert _{\zeta }C^{\theta } _{\theta \zeta ^{*}} \Vert u_{n} \Vert ^{\theta }}{\theta \Vert u_{n} \Vert ^{p}} \\ \to &0 \quad \text{as } n\to +\infty . \end{aligned}$$
(15)

In the following, we consider two opposite cases.

Case 1: \(z_{0}\not \equiv 0\). Let \(\varOmega =\{t\in \mathbb{R}||z_{0}(t)|>0 \}\). Then we can see that \(\text{meas}(\varOmega )>0\), where meas denotes the Lebesgue measure. Then there exists \(\chi >0\) such that \(\operatorname{meas}(\varLambda )>0\), where \(\varLambda =\varOmega \cap P_{\chi }\) and \(P_{\chi }=\{t\in \mathbb{R}||t| \leq \chi \}\). Since \(\|u_{n}\|\to +\infty \) as \(n\to +\infty \), we have \(|u_{n}(t)|\to +\infty \) as \(n\to +\infty \) for a.e. \(t\in \varLambda \). By \((W_{2})\), \((W_{3})\) and Fatou’s lemma, one can get

$$\begin{aligned}& \lim_{n\to +\infty } \int _{\mathbb{R}} \frac{W_{1}(t,u_{n}(t))}{ \Vert u _{n} \Vert ^{p}}\,dt \\& \quad =\lim_{n\to +\infty } \int _{\varLambda } \frac{W_{1}(t,u_{n}(t))}{ \Vert u _{n} \Vert ^{p}}\,dt+\lim _{n\to +\infty } \int _{\mathbb{R}\setminus \varLambda } \frac{W_{1}(t,u_{n}(t))}{ \Vert u_{n} \Vert ^{p}}\,dt \\& \quad \geq \lim_{n\to +\infty } \int _{\varLambda } \frac{W_{1}(t,u_{n}(t))}{ \vert u _{n}(t) \vert ^{{p}}} \bigl\vert z_{n}(t) \bigr\vert ^{p}\,dt-d_{1} \int _{\mathbb{R}\setminus \varLambda } \bigl\vert z_{n}(t) \bigr\vert ^{p}\,dt \\& \quad \geq \lim_{n\to +\infty } \int _{\varLambda } \frac{W_{1}(t,u_{n}(t))}{ \vert u _{n}(t) \vert ^{{p}}} \bigl\vert z_{n}(t) \bigr\vert ^{{p}}\,dt-d_{1}C^{p}_{p} \Vert z_{n} \Vert ^{p} \\& \quad =+\infty , \end{aligned}$$

which contradicts (15). So \(\|u_{n}\|\) is bounded in this case.

Case 2: \(z_{0}\equiv 0\). Set

$$\begin{aligned} \widetilde{W_{1}}(t,x)= \bigl(\nabla W_{1}(t,x),x \bigr)- \nu W_{1}(t,x), \end{aligned}$$

where ν is defined in \((W_{4})\). From \((W_{1})\), we can deduce that \(\widetilde{W_{1}}(t,x)=o(|x|^{{p}})\) as \(|x|\to 0\), then there exists \(\rho _{1}\in (0,\rho _{0})\) such that

$$\begin{aligned} \bigl\vert \widetilde{W_{1}}(t,x) \bigr\vert \leq \vert x \vert ^{p} \end{aligned}$$
(16)

for all \(|x|\leq \rho _{1}\), where \(\rho _{0}\) is defined in \((W_{4})\). It follows from (6), (8), (14), (16), \((W_{4})\) and \((W_{6})\) that

$$\begin{aligned} o(1) =&\frac{\nu M_{4}+M_{4}}{ \Vert u_{n} \Vert ^{p}} \\ \geq& \frac{\nu I(u_{n})-\langle I'(u_{n}),u_{n}\rangle }{ \Vert u_{n} \Vert ^{p}} \\ \geq& \biggl(\frac{\nu }{p}-1 \biggr)+\frac{1}{ \Vert u_{n} \Vert ^{p}} \int _{\mathbb{R}}\widetilde{W_{1}} \bigl(t,u_{n}(t) \bigr)\,dt-\frac{\nu +\theta }{ \theta \Vert u_{n} \Vert ^{p}} \Vert c \Vert _{\zeta }C^{\theta }_{\theta \zeta ^{*}} \Vert u _{n} \Vert ^{\theta } \\ \geq& \biggl(\frac{\nu }{{p}}-1 \biggr)+ \frac{1}{ \Vert u_{n} \Vert ^{p}} \int _{ \vert u_{n} \vert \leq \rho _{1}}\widetilde{W_{1}} \bigl(t,u_{n}(t) \bigr)\,dt+\frac{1}{ \Vert u_{n} \Vert ^{p}} \int _{\rho _{1}< \vert u_{n} \vert \leq \rho _{0}}\widetilde{W_{1}} \bigl(t,u _{n}(t) \bigr)\,dt \\ &{}+\frac{1}{ \Vert u_{n} \Vert ^{p}} \int _{ \vert u_{n} \vert > \rho _{0}}\widetilde{W_{1}} \bigl(t,u _{n}(t) \bigr)\,dt-o(1) \\ \geq &\biggl(\frac{\nu }{p}-1 \biggr)-\frac{1}{ \Vert u_{n} \Vert ^{p}} \biggl( \int _{ \vert u_{n} \vert \leq \rho _{1}} \bigl\vert u_{n}(t) \bigr\vert ^{p}\,dt+d_{2} \int _{ \vert u_{n} \vert >\rho _{0}} \bigl\vert u _{n}(t) \bigr\vert ^{p}\,dt \biggr) \\ &{}-\frac{\max_{\rho _{1}< \vert x \vert \leq \rho _{0}} \vert \widetilde{W_{1}}(t,x) \vert }{ \rho ^{p}_{1}} \int _{\rho _{1}< \vert u_{n} \vert \leq \rho _{0}}\frac{ \vert u_{n}(t) \vert ^{p}}{ \Vert u_{n} \Vert ^{p}}\,dt-o(1) \\ \geq &\biggl(\frac{\nu }{p}-1 \biggr)- \biggl(1+d_{2}+ \frac{ \max_{\rho _{1}< \vert x \vert \leq \rho _{0}} \vert \widetilde{W_{1}}(t,x) \vert }{\rho ^{p} _{1}} \biggr) \int _{\mathbb{R}} \bigl\vert z_{n}(t) \bigr\vert ^{p}\,dt-o(1) \\ \to& \frac{\nu }{p}-1\quad \text{as } n\to +\infty , \end{aligned}$$

which is a contradiction. Therefore, \(\|u_{n}\|\) is bounded.

Going if necessary to a subsequence, we can assume that \(u_{n}\rightharpoonup u\) in E, which yields

$$\begin{aligned} \bigl\langle I'(u_{n})-I'(u),u_{n}-u \bigr\rangle =& \Vert u_{n}-u \Vert ^{p} \\ &{}- \int _{\mathbb{R}} \bigl(\nabla W_{1} \bigl(t,u_{n}(t) \bigr)-\nabla W_{1} \bigl(t,u(t) \bigr),u _{n}(t)-u(t) \bigr)\,dt \\ &{}- \int _{\mathbb{R}} \bigl(\nabla W_{2} \bigl(t,u_{n}(t) \bigr)-\nabla W_{2} \bigl(t,u(t) \bigr),u _{n}(t)-u(t) \bigr)\,dt \\ \to &0 \quad \text{as}\ n\to +\infty . \end{aligned}$$
(17)

It follows from (2), \((W_{0})\) and Lemma 2 that

$$\begin{aligned}& \int _{\mathbb{R}} \bigl(\nabla W_{1} \bigl(t,u_{n}(t) \bigr)-\nabla W_{1} \bigl(t,u(t) \bigr), u _{n}(t)-u(t) \bigr)\,dt \\& \quad \leq \int _{\mathbb{R}} \bigl(c_{1} \bigl\vert u_{n}(t) \bigr\vert ^{\mu -1}+c_{1} \bigl\vert u(t) \bigr\vert ^{\mu -1}+2c _{2} \bigr) \bigl\vert u_{n}(t)-u(t) \bigr\vert \,dt \\& \quad \leq \bigl(c_{1}C^{\mu -1}_{\infty } \Vert u_{n} \Vert ^{\mu -1}+c_{1}C^{\mu -1} _{\infty } \Vert u \Vert ^{\mu -1}+2c_{2} \bigr) \Vert u_{n}-u \Vert _{1} \\& \quad \to 0 \quad \text{as } n\to +\infty . \end{aligned}$$
(18)

On account of the continuity of \(\nabla W_{2}(t,x)\) and \(u_{n}\to u\) in \(L^{\infty }_{\mathrm{loc}}(\mathbb{R},\mathbb{R}^{N})\), there exists \(n_{0}\in \mathbb{N}\) such that

$$\begin{aligned} \int _{|t|\leq T} \bigl(\nabla W_{2} \bigl(t, u_{n}(t) \bigr)-\nabla W_{2} \bigl(t,u(t) \bigr),u_{n}(t)-u(t) \bigr)\,dt< \varepsilon ,\quad \forall n\geq n_{0}, \end{aligned}$$
(19)

where T is defined in (12). In addition, by (12), \((W_{7})\) and Hölder’s inequality, we have

$$\begin{aligned}& \int _{ \vert t \vert >T} \bigl(\nabla W_{2} \bigl(t,u_{n}(t) \bigr)-\nabla W_{2} \bigl(t,u(t) \bigr), u_{n}(t)-u(t) \bigr)\,dt \\& \quad \leq \int _{ \vert t \vert >T}c(t) \bigl( \bigl\vert u_{n}(t) \bigr\vert ^{\theta -1}+ \bigl\vert u(t) \bigr\vert ^{\theta -1} \bigr) \bigl\vert u _{n}(t)-u(t) \bigr\vert \,dt \\& \quad \leq \Vert u_{n}-u \Vert _{\infty } \biggl( \int _{ \vert t \vert >T}c^{\zeta }(t)\,dt \biggr) ^{\frac{1}{\zeta }} \bigl( \Vert u_{n} \Vert ^{\theta -1}_{\zeta ^{*}(\theta -1)}+ \Vert u \Vert ^{\theta -1}_{\zeta ^{*}(\theta -1)} \bigr) \\& \quad \leq \varepsilon \Vert u_{n}-u \Vert _{\infty } \bigl( \Vert u_{n} \Vert ^{\theta -1} _{\zeta ^{*}(\theta -1)}+ \Vert u \Vert ^{\theta -1}_{\zeta ^{*}(\theta -1)} \bigr). \end{aligned}$$
(20)

Hence, by (17)–(20) we conclude that \(\|u_{n}-u\|\to 0\) as \(n\to +\infty \), which means that the \((C)\) condition is fulfilled. □

Lemma 7

Suppose that the conditions of Theorem 1hold, then there exist\(\varrho _{1}\), \(\alpha _{1}>0\)such that\(I|_{\partial B_{\varrho _{1}}} \geq \alpha _{1}\), where\(B_{\varrho _{1}}=\{u\in E: \|u\|\leq \varrho _{1}\}\).

Proof

In view of (7) and (8), for any \(u\in E\) and \(0<\varepsilon <(pC_{p}^{p})^{-1}\), we have

$$\begin{aligned} I(u) =& \frac{1}{p} \Vert u \Vert ^{p}- \int _{\mathbb{R}}W_{1}(t,u)\,dt- \int _{\mathbb{R}}W_{2}(t,u)\,dt \\ \geq &\frac{1}{p} \Vert u \Vert ^{p}-\varepsilon \int _{\mathbb{R}} \vert u \vert ^{p}\,dt-C _{\varepsilon } \int _{\mathbb{R}} \vert u \vert ^{\mu }\,dt- \frac{C^{\theta }_{ \theta \zeta ^{*}}}{\theta } \Vert c \Vert _{\zeta } \Vert u \Vert ^{\theta } \\ \geq &\frac{1}{p} \Vert u \Vert ^{p}-\varepsilon C^{p}_{p} \Vert u \Vert ^{p} -C_{\varepsilon }C^{\mu }_{\mu } \Vert u \Vert ^{\mu }-\frac{C^{\theta }_{\theta \zeta ^{*}}}{ \theta } \Vert c \Vert _{\zeta } \Vert u \Vert ^{\theta } \\ \geq & \biggl(\frac{1}{p}-\varepsilon C^{p}_{p} \biggr) \Vert u \Vert ^{p}-C _{\varepsilon }C^{\mu }_{\mu } \Vert u \Vert ^{\mu }- \frac{C^{\theta }_{\theta \zeta ^{*}}}{\theta } \Vert c \Vert _{\zeta } \Vert u \Vert ^{\theta }, \end{aligned}$$

which combined with \((W_{6})\) implies that there exist positive constants \(\varrho _{1}\) and \(\alpha _{1}\) such that \(I|_{\partial B _{\varrho _{1}}}\geq \alpha _{1}\). □

Lemma 8

Assume that the conditions of Theorem 1hold, then there exists\(v_{1}\in E\)such that\(\|v_{1}\|>\varrho _{1}\)and\(I(v_{1})\leq 0\), where\(\varrho _{1}\)is defined in Lemma 7.

Proof

We choose \(v_{0}\in C_{0}^{\infty }([-1,1], \mathbb{R}^{N})\) such that \(\|v_{0}\|=1\). For \(\beta >(p\int ^{1}_{-1}|v_{0}(t)|^{p}\,dt)^{-1}\), it follows from \((W_{2})\) that there exists \(\tau >0\) such that

$$\begin{aligned} W(t,x)\geq \beta \vert x \vert ^{{p}} \end{aligned}$$

for all \(|x|\geq \tau \). By \((W_{3})\), we get

$$\begin{aligned} W(t,x)\geq \beta \bigl( \vert x \vert ^{p}-\tau ^{p} \bigr)-d_{1}\tau ^{p} \end{aligned}$$
(21)

for all \((t,x)\in \mathbb{R}\times \mathbb{R}^{N}\). For \(\eta >0\), by (21) and \((W_{5})\) we have

$$\begin{aligned} I(\eta v_{0}) =&\frac{\eta ^{p}}{p}- \int ^{1}_{-1}W_{1} \bigl(t, \eta v_{0}(t) \bigr)\,dt- \int ^{1}_{-1}W_{2} \bigl(t, \eta v_{0}(t) \bigr)\,dt \\ \leq &\frac{\eta ^{p}}{p}- \int ^{1}_{-1}W_{1} \bigl(t, \eta v_{0}(t) \bigr)\,dt \\ \leq &\frac{\eta ^{p}}{p}- \int ^{1}_{-1}\beta \bigl\vert \eta v_{0}(t) \bigr\vert ^{p}\,dt+ \beta \int ^{1}_{-1}\tau ^{p} \,dt+d_{1} \int ^{1}_{-1}\tau ^{p}\,dt \\ \leq & \biggl(\frac{1}{p}-\beta \int ^{1}_{-1} \bigl\vert v_{0}(t) \bigr\vert ^{p}\,dt \biggr) \eta ^{p}+2\beta \tau ^{p}+2d_{1}\tau ^{p}, \end{aligned}$$

which implies that

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

Therefore, there exists \(\eta _{0}>0\) such that \(I(\eta _{0} v_{0})<0\). Let \(v_{1}=\eta _{0} v_{0}\), we can see \(I(v_{1})<0\), which proves this lemma. □

Proof of Theorem 1

By Lemmas 48, we can see that I possesses at least one nontrivial critical point. Then the critical point is the first homoclinic solution to (1). To get the second solution, we just need to prove that \(\inf_{u\in B_{\varrho _{1}}} I(u)<0\), where \(B_{\varrho _{1}}\) is defined in Lemma 7. We choose \(v_{2}\in C^{\infty }_{0}([-1,1], \mathbb{R}^{N})\setminus \{0\}\). Then, by \((W_{3})\) and \((W_{5})\), for any \(l>0\) we get

$$\begin{aligned} I(lv_{2}) =& \frac{l^{p}}{p} \Vert v_{2} \Vert ^{p}- \int ^{1}_{-1} W_{1} \bigl(t,lv _{2}(t) \bigr)\,dt- \int ^{1}_{-1} W_{2} \bigl(t,lv_{2}(t) \bigr)\,dt \\ \leq &\frac{l^{p}}{p} \Vert v_{2} \Vert ^{p}+d_{1}l^{p} \int ^{1}_{-1} \bigl\vert v_{2}(t) \bigr\vert ^{p}\,dt-l ^{\theta } \int ^{1}_{-1}b(t) \bigl\vert v_{2}(t) \bigr\vert ^{\theta }\,dt \\ \leq &\frac{l^{p}}{p} \Vert v_{2} \Vert ^{p}+d_{1}l^{p} \int ^{1}_{-1} \bigl\vert v_{2}(t) \bigr\vert ^{p}\,dt-l ^{\theta } \Bigl(\min _{t\in [-1,1]}b(t) \Bigr) \int ^{1}_{-1} \bigl\vert v_{2}(t) \bigr\vert ^{ \theta }\,dt \\ < &0 \end{aligned}$$

for l small enough, which implies that \(\delta _{1}= \inf_{u\in B_{\varrho _{1}}} I(u)<0\). Then it follows from Ekeland’s variational principle that there exists a minimizing sequence \(\{v_{n}\}\subset B_{\varrho _{1}}\) such that

$$\begin{aligned} \delta _{1}\leq I(v_{n})< \delta _{1}+ \frac{1}{n} \quad \text{and} \quad I(u)\geq I(v_{n})- \frac{1}{n} \Vert u-v_{n} \Vert \quad \text{for } u \in B_{\varrho _{1}}. \end{aligned}$$

Thus, \(\{v_{n}\}\) is a bounded \((PS)\) sequence, which means that it is also a (C) sequence. Then from Lemma 6, there exists \(u_{1}\in E\) such that \(I'(u_{1})=0\) and \(I(u_{1})<0\). In conclusion, problem (1) possesses at least two nontrivial homoclinic solutions. □

Proof of Theorem 2

In this section, we still work in the Banach space

$$\begin{aligned} E= \biggl\{ u\in W^{1,p} \bigl(\mathbb{R}, \mathbb{R}^{N} \bigr): \int _{\mathbb{R}} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p}+a(t) \bigl\vert u(t) \bigr\vert ^{p} \bigr) \,dt< +\infty \biggr\} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u \Vert = \biggl( \int _{\mathbb{R}} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p}+a(t) \bigl\vert u(t) \bigr\vert ^{p} \bigr) \,dt \biggr) ^{\frac{1}{p}}. \end{aligned}$$

Lemma 9

Suppose that the condition\((A_{2})\)holds, the embedding\(E\hookrightarrow L^{1}(\mathbb{R}, \mathbb{R}^{N})\)is continuous and compact.

Proof

Assume that \(\{u_{n}\}\subset E\) such that \(u_{n}\rightharpoonup 0\) in E. We will show that \(u_{n}\to 0\) in \(L_{1}(\mathbb{R}, \mathbb{R} ^{N})\). By the Banach–Steinhaus theorem, there exists \(M_{5}>0\) such that

$$\begin{aligned} \sup_{n\in \mathbb{N}} \Vert u_{n} \Vert \leq M_{5}. \end{aligned}$$

For any \(\varepsilon >0\), by condition (\(A_{2}\)) there is \(r_{0}>0\) such that

$$\begin{aligned} \operatorname{meas} B_{\varepsilon }< \varepsilon , \end{aligned}$$

where

$$\begin{aligned} B_{\varepsilon }= \bigl\{ t\in \mathbb{R}\setminus (-r_{0}, r_{0}) |\ \vert t \vert ^{-\lambda p} a(t)< \varepsilon ^{-1} \bigr\} . \end{aligned}$$

Let

$$\begin{aligned} D_{\varepsilon } &=\mathbb{R}\setminus \bigl((-r_{0}, r_{0})\cup B _{\varepsilon } \bigr), \\ \mu _{\varepsilon } &=\inf_{ t\in D_{\varepsilon }} \vert t \vert ^{-\lambda p}a(t), \end{aligned}$$

then \(\frac{1}{\mu _{\varepsilon }}\leq \varepsilon \). On the one hand, one has

$$\begin{aligned} \int _{ \vert t \vert \geq r_{0}} \vert u_{n} \vert \,dt =& \int _{{B_{\varepsilon }}} \vert u_{n} \vert \,dt+ \int _{{D_{\varepsilon }}} \vert u_{n} \vert \,dt \\ \leq & \Vert u_{n} \Vert _{\infty }\cdot \text{meas}\ {B_{\varepsilon }}+ \int _{D_{\varepsilon }} \vert t \vert ^{\lambda } \vert u_{n} \vert \vert t \vert ^{-\lambda } \,dt \\ \leq &\varepsilon C_{\infty }M_{5}+ \biggl( \int _{D_{\varepsilon }} \vert t \vert ^{ \lambda p} \vert u_{n} \vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{ \vert t \vert \geq r _{0}} \vert t \vert ^{-\lambda q}\,dt \biggr)^{\frac{1}{q}} \\ \leq & \varepsilon C_{\infty }M_{5}+\delta _{2} \mu _{\varepsilon }^{- \frac{1}{p}} \biggl( \int _{D_{\varepsilon }}a(t) \vert u_{n} \vert ^{p}\,dt \biggr) ^{\frac{1}{p}} \\ \leq &\varepsilon C_{\infty }M_{5}+\varepsilon ^{\frac{1}{p}}\delta _{2}M_{5}, \end{aligned}$$
(22)

where \(\delta _{2}= (\int _{|t|\geq r_{0}}|t|^{-\lambda q}\,dt ) ^{\frac{1}{q}}\). On the other hand, it follows from the Sobolev compact embedding theorem that \(u_{n}\to 0\) in \(L^{1}((-r_{0},r_{0}), \mathbb{R}^{N})\). Therefore, the embedding \(E\hookrightarrow L^{1}( \mathbb{R},\mathbb{R}^{N})\) is compact.

Now for \(\varepsilon =1\), by (22) we have

$$\begin{aligned} \int _{ \vert t \vert \geq r_{0}} \vert u \vert \,dt\leq C_{\infty } \Vert u \Vert +\delta _{2} \Vert u \Vert = (C _{\infty }+\delta _{2}) \Vert u \Vert ,\quad \forall u\in E, \end{aligned}$$

which implies that the embedding is also continuous. □

Proof of Theorem 2

By similar steps to the proof of Theorem 1, we can obtain the conclusion of Theorem 2. □

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Wan, L. Multiple homoclinic solutions for p-Laplacian Hamiltonian systems with concave–convex nonlinearities. Bound Value Probl 2020, 4 (2020). https://doi.org/10.1186/s13661-019-01317-z

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MSC

  • 34C37
  • 35A15
  • 35B38

Keywords

  • Homoclinic solutions
  • p-Laplacian operator
  • Variational methods