Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential

Abstract

In the present paper, we deal with the existence and multiplicity of homoclinic solutions of the second-order self-adjoint discrete Hamiltonian system

\mathrm{△}[p(n)\mathrm{△}u(n-1)]-L(n)u(n)+\mathrm{\nabla }W(n,u(n))=0.

Under the assumption that W(n,x) is of indefinite sign and subquadratic as |x|\to +\mathrm{\infty } and p(n) and L(n) are \mathcal{N}\times \mathcal{N} real symmetric positive definite matrices for all n\in \mathbb{Z}, and that

\underset{|n|\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\left[{|n|}^{\nu -2}\underset{|x|=1}{inf}(L(n)x,x)\right]>0

for some constant , we establish some existence criteria to guarantee that the above system has at least one or multiple homoclinic solutions by using Clark’s theorem in critical point theory.

MSC:39A11, 58E05, 70H05.

1 Introduction

Consider the second-order self-adjoint discrete Hamiltonian system

\mathrm{△}[p(n)\mathrm{△}u(n-1)]-L(n)u(n)+\mathrm{\nabla }W(n,u(n))=0,

(1.1)

where n\in \mathbb{Z}, u\in {\mathbb{R}}^{\mathcal{N}}, \mathrm{△}u(n)=u(n+1)-u(n) is the forward difference operator, p,L:\mathbb{Z}\to {\mathbb{R}}^{\mathcal{N}\times \mathcal{N}} and W:\mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}}\to \mathbb{R}. As usual, we say that a solution u(n) of system (1.1) is homoclinic (to 0) if u(n)\to 0 as n\to \pm \mathrm{\infty }. In addition, if u(n)\not\equiv 0, then u(n) is called a nontrivial homoclinic solution.

In general, system (1.1) may be regarded as a discrete analogue of the following second order Hamiltonian system:

\frac{d}{dt}(p(t)\dot{u}(t))-L(t)u(t)+\mathrm{\nabla }W(t,u(t))=0.

(1.2)

Moreover, system (1.1) does have its applicable setting as evidenced by monographs [1, 2]. System (1.2) can also be regarded as a special form of the Emden-Fowler equation appearing in the study of astrophysics, gas dynamics, fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting system, and many well-known results concerning properties of solutions of (1.2) are collected in [3].

In papers [4–8], the authors studied the existence of homoclinic solutions of system (1.1) or some of its special forms under the following superquadratic growth (AR)-condition on W: there is a constant \mu >2 such that

or other superquadratic growth conditions, where and in the sequel, (\cdot ,\cdot ) denotes the standard inner product in {\mathbb{R}}^{\mathcal{N}} and |\cdot | is the induced norm.

When W(n,x) is of subquadratic growth at infinity, Tang and Lin [9] recently established the following results on the existence of homoclinic solutions of system (1.1).

Theorem A [9]

Assume that p(n) is an \mathcal{N}\times \mathcal{N} real symmetric positive definite matrix for all n\in \mathbb{Z}, and that L and W satisfy the following assumptions:

• (L) L(n) is an \mathcal{N}\times \mathcal{N} real symmetric positive definite matrix for all n\in \mathbb{Z}, and there exists a constant \beta >0 such that

  (L(n)x,x)\ge \beta {|x|}^2,\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}};

(W1) For every n\in \mathbb{Z}, W is continuously differentiable in x, and there exist two constants and two functions a_1,a_2\in l^{2/(2-{\gamma }_1)}(\mathbb{Z},[0,+\mathrm{\infty })) such that

|W(n,x)|\le a_1(n){|x|}^{{\gamma }_1},\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}},|x|\le 1

and

|W(n,x)|\le a_2(n){|x|}^{{\gamma }_2},\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}},|x|\ge 1;

(W2) There exist two functions b\in l^{2/(2-{\gamma }_1)}(\mathbb{Z},[0,+\mathrm{\infty })) and \phi \in C([0,+\mathrm{\infty }),[0,+\mathrm{\infty })) such that

|\mathrm{\nabla }W(n,x)|\le b(n)\phi (|x|),\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}},

where \phi (s)=O(s^{{\gamma }_1-1}) as s\to 0^{+};

(W3) There exist an n_0\in \mathbb{Z} and two constants \eta >0 and {\gamma }_3\in (1,2) such that

W(n_0,x)\ge \eta {|x|}^{{\gamma }_3},\mathrm{\forall }x\in {\mathbb{R}}^{\mathcal{N}},|x|\le 1.

Then system (1.1) possesses at least one nontrivial homoclinic solution.

Theorem B [9]

Assume that p(n) is an \mathcal{N}\times \mathcal{N} real symmetric positive definite matrix for all n\in \mathbb{Z}, and that L and W satisfy (L), (W1), (W2) and the following assumptions:

(W4) There exist two constants \eta >0 and {\gamma }_3\in (1,2) and a set J\subset \mathbb{Z} with m>0 elements such that

W(n,x)\ge \eta {|x|}^{{\gamma }_3},\mathrm{\forall }(n,x)\in J\times {\mathbb{R}}^{\mathcal{N}},|x|\le 1;

(W5) W(n,-x)=W(n,x), \mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}}.

Then system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions.

When L(n) satisfies (L) in Theorems A and B, assumption (W1) is optimal in some sense, essentially, the summable functions a_1,a_2\in l^{2/(2-{\gamma }_1)}(\mathbb{R},[0,+\mathrm{\infty })) are necessary; see Lemma 2.2 in Section 2.

Now a natural question is whether the conditions on the potential W(n,x) can be further relaxed when one imposes stronger conditions on L(n)?

In the present paper, we give a positive answer to the above question. In fact, we employ Clark’s theorem in critical point theory to establish new existence criteria to guarantee that system (1.1) has at least one or multiple homoclinic solutions under the following assumption instead of (L):

(L _ν ) L(n) is an \mathcal{N}\times \mathcal{N} real symmetric positive definite matrix for all n\in \mathbb{Z}, and there exists a constant such that

\underset{|n|\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\left[{|n|}^{\nu -2}\underset{|x|=1}{inf}(L(n)x,x)\right]>0.

Our main results are the following two theorems.

Theorem 1.1 Assume that p(n) is an \mathcal{N}\times \mathcal{N} real symmetric positive definite matrix for all n\in \mathbb{Z}, that L satisfies (L _ν ) and W satisfies the following assumptions:

(W1′) There exist constants and a_1,a_2\ge 0 such that

|W(n,x)|\le a_1{|x|}^{{\gamma }_1}+a_2{|x|}^{{\gamma }_2},\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}};

(W2′) There exists a function \phi \in C([0,+\mathrm{\infty }),[0,+\mathrm{\infty })) such that

|\mathrm{\nabla }W(n,x)|\le \phi (|x|),\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}},

where \phi (s)=O(s^{{\gamma }_3-1}) as s\to 0^{+}, ;

(W3′) There exist n_0\in \mathbb{Z} and constants {\delta }_0,\eta >0 and such that

W(n_0,x)\ge \eta {|x|}^{{\gamma }_4},\mathrm{\forall }x\in {\mathbb{R}}^{\mathcal{N}},|x|\le {\delta }_0.

Then system (1.1) possesses at least one nontrivial homoclinic solution.

Theorem 1.2 Assume that p(n) is an \mathcal{N}\times \mathcal{N} real symmetric positive definite matrix for all n\in \mathbb{Z}, and that L and W satisfy (L _ν ), (W1′), (W2′), (W5) and the following assumption:

(W4′) There exist constants {\delta }_0,\eta >0 and and a set J\subset \mathbb{Z} with m>0 elements such that

W(n,x)\ge \eta {|x|}^{{\gamma }_4},\mathrm{\forall }(n,x)\in J\times {\mathbb{R}}^{\mathcal{N}},|x|\le {\delta }_0.

Then system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions.

Remark 1.3 Obviously, assumptions (W1′), (W2′), (W3′) and (W4′) are weaker than (W1), (W2), (W3) and (W4), respectively.

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proofs of our theorems. In Section 4, we give some examples to illustrate our results.

2 Preliminaries

In this section, we always assume that p(n) and L(n) are real symmetric positive definite matrices for all n\in \mathbb{Z}. Let

and for u,v\in E, let

〈u,v〉=\sum _{n\in \mathbb{Z}}[(p(n+1)\mathrm{△}u(n),\mathrm{△}v(n))+(L(n)u(n),v(n))].

Then E is a Hilbert space with the above inner product, and the corresponding norm is

\parallel u\parallel ={\left\{\sum _{n\in \mathbb{Z}}[(p(n+1)\mathrm{△}u(n),\mathrm{△}u(n))+(L(n)u(n),u(n))]\right\}}^{1/2},u\in E.

As usual, for , set

and

and their norms are defined by

\begin{array}{c}{\parallel u\parallel }_q={\left(\sum _{n\in \mathbb{Z}}{|u(n)|}^q\right)}^{1/q},\mathrm{\forall }u\in l^q(\mathbb{Z},{\mathbb{R}}^{\mathcal{N}});\hfill \\ {\parallel u\parallel }_{\mathrm{\infty }}=\underset{n\in \mathbb{Z}}{sup}|u(n)|,\mathrm{\forall }u\in l^{\mathrm{\infty }}(\mathbb{Z},{\mathbb{R}}^{\mathcal{N}}),\hfill \end{array}

respectively.

Lemma 2.1 [9]

For u\in E,

{\parallel u\parallel }_{\mathrm{\infty }}\le \frac{1}{\sqrt[4]{(\beta +4\alpha )\beta }}\parallel u\parallel ,

(2.1)

where \alpha =inf\{(p(n)x,x):n\in \mathbb{Z},x\in {\mathbb{R}}^{\mathcal{N}},|x|=1\} and \beta =inf\{(L(n)x,x):n\in \mathbb{Z},x\in {\mathbb{R}}^{\mathcal{N}},|x|=1\}.

By (L _ν ), there exist two constants N_0>0 and M_0>0 such that

{|n|}^{\nu -2}\underset{|x|=1}{inf}(L(n)x,x)\ge M_0,|n|\ge N_0,

which implies

{|n|}^{\nu -2}(L(n)x,x)\ge M_0{|x|}^2,|n|\ge N_0,x\in {\mathbb{R}}^{\mathcal{N}}.

(2.2)

Lemma 2.2 Suppose that L satisfies (L _ν ). Then, for 1\le q\in (2/(3-\nu ),2), E is compactly embedded in l^q(\mathbb{Z},{\mathbb{R}}^{\mathcal{N}}); moreover,

\sum _{|n|>N}{|u(n)|}^q\le \frac{K(q)}{N^{\kappa }}{\parallel u\parallel }^q,\mathrm{\forall }u\in E,N\ge N_0

(2.3)

and

{\parallel u\parallel }_q^q\le [{\left(\sum _{|n|\le N}{[l(n)]}^{-q/(2-q)}\right)}^{1-\frac{q}{2}}+\frac{K(q)}{N^{\kappa }}]{\parallel u\parallel }^q,\mathrm{\forall }u\in E,N\ge N_0,

(2.4)

where

\kappa =\frac{(3-\nu )q-2}{2}>0,K(q)={\left[\frac{2(2-q)}{(3-\nu )q-2}\right]}^{1-\frac{q}{2}}{M}_0^{-q/2}

(2.5)

and

l(n)=\underset{x\in {\mathbb{R}}^{\mathcal{N}},|x|=1}{inf}(L(n)x,x).

(2.6)

Proof Let r=[(3-\nu )q-2]/(2-q). Then r>0. For u\in E and N\ge N_0, it follows from (2.2), (2.5) and the Hölder inequality that

\begin{array}{rcl}\sum _{|n|>N}{|u(n)|}^q& \le & {\left(\sum _{|n|>N}{|n|}^{-(2-\nu )q/(2-q)}\right)}^{1-\frac{q}{2}}{\left(\sum _{|n|>N}{|n|}^{2-\nu }{|u(n)|}^2\right)}^{\frac{q}{2}}\\ \le & {\left(\frac{2}{r{N}^r}\right)}^{1-\frac{q}{2}}{\left[\frac{1}{M_0}\sum _{|n|>N}(L(n)u(n),u(n))\right]}^{\frac{q}{2}}\\ \le & \frac{2^{(2-q)/2}}{M_0^{q/2}{r}^{(2-q)/2}{N}^{\kappa }}{\parallel u\parallel }^q=\frac{K(q)}{N^{\kappa }}{\parallel u\parallel }^q.\end{array}

This shows that (2.3) holds. Hence, from (2.2), (2.6) and the Hölder inequality, one has

\begin{array}{rcl}{\parallel u\parallel }_q^q& =& \sum _{|n|\le N}{|u(n)|}^q+\sum _{|n|>N}{|u(n)|}^q\\ \le & {\left(\sum _{|n|\le N}{[l(n)]}^{-q/(2-q)}\right)}^{1-\frac{q}{2}}{(\sum _{|n|\le N}l(n){|u(n)|}^2)}^{\frac{q}{2}}+\frac{K(q)}{N^{\kappa }}{\parallel u\parallel }^q\\ \le & {\left(\sum _{|n|\le N}{[l(n)]}^{-q/(2-q)}\right)}^{1-\frac{q}{2}}{\parallel u\parallel }^q+\frac{K(q)}{N^{\kappa }}{\parallel u\parallel }^q.\end{array}

This shows that (2.4) holds.

Finally, we prove that E is compactly embedded in l^q(\mathbb{Z},{\mathbb{R}}^{\mathcal{N}}). Let \{u_k\}\subset E be a bounded sequence. Then by (2.1), there exists a constant \mathrm{\Lambda }>0 such that

{\parallel u_k\parallel }_{\mathrm{\infty }}\le \frac{1}{\sqrt[4]{(\beta +4\alpha )\beta }}\parallel u_k\parallel \le \mathrm{\Lambda },k\in \mathbb{N}.

(2.7)

Since E is reflexive, \{u_k\} possesses a weakly convergent subsequence in E. Passing to a subsequence if necessary, it can be assumed that u_k\rightharpoonup u_0 in E. It is easy to verify that

\underset{k\to \mathrm{\infty }}{lim}{u}_k(n)=u_0(n),\mathrm{\forall }n\in \mathbb{Z}.

(2.8)

For any given number \epsilon >0, we can choose N_{\epsilon }>0 such that

(2.9)

It follows from (2.8) that there exists k_0\in \mathbb{N} such that

(2.10)

On the other hand, it follows from (2.3), (2.7) and (2.9) that

\begin{array}{rcl}\sum _{|n|>N_{\epsilon }}{|u_k(n)-u_0(n)|}^q& \le & 2^{q-1}\sum _{|n|>N_{\epsilon }}({|u_k(n)|}^q+{|u_0(n)|}^q)\\ \le & \frac{2^{q-1}K(q)}{N_{\epsilon }^{\kappa }}({\parallel u_k\parallel }^q+{\parallel u_0\parallel }^q)\\ \le & \frac{2^{q-1}K(q)}{N_{\epsilon }^{\kappa }}\{{\left[\sqrt[4]{(\beta +4\alpha )\beta }\mathrm{\Lambda }\right]}^q+{\parallel u_0\parallel }^q\}\le \epsilon ,k\in \mathbb{N}.\end{array}

(2.11)

Since ε is arbitrary, combining (2.10) with (2.11), we get

{\parallel u_k-u_0\parallel }_q^q=\sum _{n\in \mathbb{Z}}{|u_k(n)-u_0(n)|}^q\to 0\text{as }k\to +\mathrm{\infty }.

This shows that \{u_k\} possesses a convergent subsequence in l^q(\mathbb{Z},{\mathbb{R}}^{\mathcal{N}}). Therefore, E is compactly embedded in l^q(\mathbb{Z},{\mathbb{R}}^{\mathcal{N}}) for 1\le q\in (2/(3-\nu ),2). □

Lemma 2.3 Suppose that L and W satisfy (L _ν ) and (W1′). Then, for u\in E,

\sum _{n\in \mathbb{Z}}\left|W(n,u(n))\right|\le {\varphi }_1(N){\parallel u\parallel }^{{\gamma }_1}+{\varphi }_2(N){\parallel u\parallel }^{{\gamma }_2},N\ge N_0,

(2.12)

where

\begin{array}{r}{\kappa }_1=\frac{(3-\nu ){\gamma }_1-2}{2},\\ {\kappa }_2=\frac{(3-\nu ){\gamma }_2-2}{2};\end{array}

(2.13)

{\varphi }_1(N)=a_1[{\left(\sum _{|n|\le N}{[l(n)]}^{-{\gamma }_1/(2-{\gamma }_1)}\right)}^{1-\frac{{\gamma }_1}{2}}+\frac{K({\gamma }_1)}{N^{{\kappa }_1}}],

(2.14)

{\varphi }_2(N)=a_2[{\left(\sum _{|n|\le N}{[l(n)]}^{-{\gamma }_2/(2-{\gamma }_2)}\right)}^{1-\frac{{\gamma }_2}{2}}+\frac{K({\gamma }_2)}{N^{{\kappa }_2}}].

(2.15)

Proof For N\ge N_0, it follows from (2.4) and (W1′) that

\begin{array}{rcl}\sum _{n\in \mathbb{Z}}\left|W(n,u(n))\right|& \le & a_1\sum _{n\in \mathbb{Z}}{|u(n)|}^{{\gamma }_1}+a_2\sum _{n\in \mathbb{Z}}{|u(n)|}^{{\gamma }_2}\\ \le & a_1[{\left(\sum _{|n|\le N}{[l(n)]}^{-{\gamma }_1/(2-{\gamma }_1)}\right)}^{1-\frac{{\gamma }_1}{2}}+\frac{K({\gamma }_1)}{N^{{\kappa }_1}}]{\parallel u\parallel }^{{\gamma }_1}\\ +a_2[{\left(\sum _{|n|\le N}{[l(n)]}^{-{\gamma }_2/(2-{\gamma }_2)}\right)}^{1-\frac{{\gamma }_2}{2}}+\frac{K({\gamma }_2)}{N^{{\kappa }_2}}]{\parallel u\parallel }^{{\gamma }_2}\\ =& {\varphi }_1(N){\parallel u\parallel }^{{\gamma }_1}+{\varphi }_2(N){\parallel u\parallel }^{{\gamma }_2}.\end{array}

This shows (2.12) holds. □

Lemma 2.4 Assume that L and W satisfy (L _ν ), (W1′) and (W2′). Then the functional f:E\to \mathbb{R} defined by

f(u)=\frac{1}{2}{\parallel u\parallel }^2-\sum _{n\in \mathbb{Z}}W(n,u(n)),\mathrm{\forall }u\in E

(2.16)

is well defined and of class C^1(E,\mathbb{R}) and

\begin{array}{rcl}〈f^{\prime }(u),v〉& =& \sum _{n\in \mathbb{Z}}[(p(n+1)\mathrm{△}u(n),\mathrm{△}v(n))+(L(n)u(n),v(n))\\ -(\mathrm{\nabla }W(n,u(n)),v(n))],\mathrm{\forall }u,v\in E.\end{array}

(2.17)

Furthermore, the critical points of f in E are solutions of (1.1) with u(\pm \mathrm{\infty })=0.

Proof Lemma 2.3 implies that f defined by (2.16) is well defined on E. Next, we prove that (2.17) holds. By (W2′), one can choose an M_1>0 such that

\phi (|x|)\le M_1{|x|}^{{\gamma }_3-1},\mathrm{\forall }x\in {\mathbb{R}}^{\mathcal{N}},|x|\le 1.

(2.18)

For any u,v\in E, there exists an integer N_1>N_0 such that for |n|>N_1. Then, for any sequence {\{{\theta }_n\}}_{n\in \mathbb{Z}}\subset \mathbb{R} with for n\in \mathbb{Z} and any number h\in (0,1), by (W2′), (2.3) and Lemma 2.2, we have

(2.19)

where {\kappa }_3=[{\gamma }_3(3-\nu )-2]/2>0. Then by (2.16), (2.19) and Lebesgue’s dominated convergence theorem, we have

\begin{array}{rcl}〈f^{\prime }(u),v〉& =& \underset{h\to 0^{+}}{lim}\frac{f(u+hv)-f(u)}{h}\\ =& \underset{h\to 0^{+}}{lim}[〈u,v〉+\frac{h{\parallel v\parallel }^2}{2}-\sum _{n\in \mathbb{Z}}(\mathrm{\nabla }W(n,u(n)+{\theta }_{n}hv(n)),v(n))]\\ =& 〈u,v〉-\sum _{n\in \mathbb{Z}}(\mathrm{\nabla }W(n,u(n)),v(n))\\ =& \sum _{n\in \mathbb{Z}}[(p(n+1)\mathrm{△}u(n),\mathrm{△}v(n))+(L(n)u(n),v(n))-(\mathrm{\nabla }W(n,u(n)),v(n))].\end{array}

This shows that (2.17) holds. Observe that for u,v\in E,

\begin{array}{c}\sum _{n\in \mathbb{Z}}[(p(n+1)\mathrm{△}u(n),\mathrm{△}v(n))+(L(n)u(n),v(n))-(\mathrm{\nabla }W(n,u(n)),v(n))]\hfill \\ =\sum _{n\in \mathbb{Z}}[\mathrm{△}(p(n)\mathrm{△}u(n-1),v(n))-(\mathrm{△}(p(n)\mathrm{△}u(n-1)),v(n))+(L(n)u(n),v(n))\hfill \\ -(\mathrm{\nabla }W(n,u(n)),v(n))]\hfill \\ =\sum _{n\in \mathbb{Z}}\left[(-\mathrm{△}(p(n)\mathrm{△}u(n-1))+L(n)u(n)-\mathrm{\nabla }W(n,u(n)),v(n))\right].\hfill \end{array}

(2.20)

It follows from (2.17) and (2.20) that 〈f^{\prime }(u),v〉=0 for all v\in E if and only if

\mathrm{△}(p(n)\mathrm{△}u(n-1))-L(n)u(n)+\mathrm{\nabla }W(n,u(n))=0,\mathrm{\forall }n\in \mathbb{Z}.

So, the critical points of f in E are the solutions of system (1.1) with u(\pm \mathrm{\infty })=0.

Let us prove now that f^{\prime } is continuous. Let u_k\to u in E. Then there exists a constant \delta >0 such that

\parallel u\parallel \le \sqrt[4]{(\beta +4\alpha )\beta }\delta ,\parallel u_k\parallel \le \sqrt[4]{(\beta +4\alpha )\beta }\delta ,k=1,2,\dots .

(2.21)

It follows from (2.1) that

{\parallel u\parallel }_{\mathrm{\infty }}\le \delta ,{\parallel u_k\parallel }_{\mathrm{\infty }}\le \delta ,k=1,2,\dots .

(2.22)

By (W2′), one can choose an M_2>0 such that

\phi (|x|)\le M_2{|x|}^{{\gamma }_3-1},\mathrm{\forall }x\in {\mathbb{R}}^{\mathcal{N}},|x|\le \delta .

(2.23)

From (2.3), (2.17), (2.21), (2.22), (2.23), (W2′) and the Hölder inequality, we have

\begin{array}{c}\left|〈f^{\prime }(u_k)-f^{\prime }(u),v〉\right|\hfill \\ \le \left|\sum _{n\in \mathbb{Z}}[(p(n+1)(\mathrm{△}{u}_k(n)-\mathrm{△}u(n)),\mathrm{△}v(n))+(L(n)(u_k(n)-u(n)),v(n))]\right|\hfill \\ +\sum _{n\in \mathbb{Z}}\left|(\mathrm{\nabla }W(n,u_k(n))-\mathrm{\nabla }W(n,u(n)),v(n))\right|\hfill \\ =|〈u_k-u,v〉|+\sum _{n\in \mathbb{Z}}\left|(\mathrm{\nabla }W(n,u_k(n))-\mathrm{\nabla }W(n,u(n)),v(n))\right|\hfill \\ \le \parallel u_k-u\parallel \parallel v\parallel +\sum _{|n|\le N}|\mathrm{\nabla }W(n,u_k(n))-\mathrm{\nabla }W(n,u(n))||v(n)|\hfill \\ +\sum _{|n|>N}(\left|\mathrm{\nabla }W(n,u_k(n))\right|+\left|\mathrm{\nabla }W(n,u(n))\right|)|v(n)|\hfill \\ \le o(1)+M_2\sum _{|n|>N}({|u_k(n)|}^{{\gamma }_3-1}+{|u(n)|}^{{\gamma }_3-1})|v(n)|\hfill \\ \le o(1)+M_2{\left(\sum _{|n|>N}{|u_k(n)|}^{{\gamma }_3}\right)}^{1-\frac{1}{{\gamma }_3}}{\left(\sum _{|n|>N}{|v(n)|}^{{\gamma }_3}\right)}^{\frac{1}{{\gamma }_3}}\hfill \\ +M_2{\left(\sum _{|n|>N}{|u(n)|}^{{\gamma }_3}\right)}^{1-\frac{1}{{\gamma }_3}}{\left(\sum _{|n|>N}{|v(n)|}^{{\gamma }_3}\right)}^{\frac{1}{{\gamma }_3}}\hfill \\ \le o(1)+\frac{M_{2}K({\gamma }_3)}{N^{{\kappa }_3}}({\parallel u_k\parallel }^{{\gamma }_3-1}+{\parallel u\parallel }^{{\gamma }_3-1})\parallel v\parallel \hfill \\ =o(1),k\to +\mathrm{\infty },N\to +\mathrm{\infty },\mathrm{\forall }v\in E,\hfill \end{array}

which implies the continuity of f^{\prime }. The proof is complete. □

Lemma 2.5 [10]

Let E be a real Banach space and let f\in C^1(E,\mathbb{R}) satisfy the (PS)-condition. If f is bounded from below, then c={inf}_{E}f is a critical value of f.

Lemma 2.6 [11]

Let E be a real Banach space, let f\in C^1(E,\mathbb{R}) with f even, bounded from below, and satisfying the (PS)-condition. Suppose that f(0)=0, there is a set K\subset E such that K is homeomorphic to S^{k-1} by an odd map, and . Then f possesses at least k distinct pairs of critical points.

3 Proofs of theorems

Proof of Theorem 1.1 In view of Lemma 2.4, f\in C^1(E,\mathbb{R}). In what follows, we first show that f is bounded from below. Choose N_2>N_0, it follows from (2.12) that

\sum _{n\in \mathbb{Z}}W(n,u(n))\le {\varphi }_1(N_2){\parallel u\parallel }^{{\gamma }_1}+{\varphi }_2(N_2){\parallel u\parallel }^{{\gamma }_2},\mathrm{\forall }u\in E.

(3.1)

By (2.16) and (3.1), we have

f(u)=\frac{1}{2}{\parallel u\parallel }^2-\sum _{n\in \mathbb{Z}}W(n,u(n))\ge \frac{1}{2}{\parallel u\parallel }^2-{\varphi }_1(N_2){\parallel u\parallel }^{{\gamma }_1}-{\varphi }_2(N_2){\parallel u\parallel }^{{\gamma }_2}.

(3.2)

Since , (3.2) implies that f(u)\to +\mathrm{\infty } as \parallel u\parallel \to +\mathrm{\infty }. Consequently, f is bounded from below.

Next, we prove that f satisfies the (PS)-condition. Assume that {\{u_k\}}_{k\in \mathbb{N}}\subset E is a sequence such that {\{f(u_k)\}}_{k\in \mathbb{N}} is bounded and f^{\prime }(u_k)\to 0 as k\to +\mathrm{\infty }. Then by (2.1) and (3.2), there exists a constant A>0 such that

{\parallel u_k\parallel }_{\mathrm{\infty }}\le \frac{1}{\sqrt[4]{(\beta +4\alpha )\beta }}\parallel u_k\parallel \le A,k\in \mathbb{N}.

(3.3)

So, passing to a subsequence if necessary, it can be assumed that u_k\rightharpoonup u_0 in E. It is easy to verify that

\underset{k\to +\mathrm{\infty }}{lim}{u}_k(n)=u_0(n),\mathrm{\forall }n\in \mathbb{Z}.

(3.4)

Hence, we have by (3.3) and (3.4)

{\parallel u_0\parallel }_{\mathrm{\infty }}\le A.

(3.5)

By virtue of (W2′), one can choose a M_3>0 such that

\phi (|x|)\le M_3{|x|}^{{\gamma }_3-1},\mathrm{\forall }x\in {\mathbb{R}}^{\mathcal{N}},|x|\le A.

(3.6)

For any given number \epsilon >0, we can choose an integer N_3>N_0 such that

(3.7)

It follows from (3.4) and the continuity of \mathrm{\nabla }W(n,x) on x that there exists k_0\in \mathbb{N} such that

(3.8)

On the other hand, it follows from (3.3), (3.5), (3.6), (3.7) and (W2′) that

\begin{array}{c}\sum _{|n|>N_3}|\mathrm{\nabla }W(n,u_k(n))-\mathrm{\nabla }W(n,u_0(n))||u_k(n)-u_0(n)|\hfill \\ \le \sum _{|n|>N_3}[\phi \left(|u_k(n)|\right)+\phi \left(|u_0(n)|\right)](|u_k(n)|+|u_0(n)|)\hfill \\ \le M_3\sum _{|n|>N_3}({|u_k(n)|}^{{\gamma }_3-1}+{|u_0(n)|}^{{\gamma }_3-1})(|u_k(n)|+|u_0(n)|)\hfill \\ \le 2M_3\sum _{|n|>N_3}({|u_k(n)|}^{{\gamma }_3}+{|u_0(n)|}^{{\gamma }_3})\hfill \\ \le \frac{2M_{3}K({\gamma }_3)}{N_3^{{\kappa }_3}}({\parallel u_k\parallel }^{{\gamma }_3}+{\parallel u_0\parallel }^{{\gamma }_3})\hfill \\ \le \frac{2M_{3}K({\gamma }_3)}{N_3^{{\kappa }_3}}\{{\left[\sqrt[4]{(\beta +4\alpha )\beta }A\right]}^{{\gamma }_3}+{\parallel u_0\parallel }^{{\gamma }_3}\}\hfill \\ \le 2M_3\epsilon ,k\in \mathbb{N}.\hfill \end{array}

(3.9)

Since ε is arbitrary, combining (3.8) with (3.9), we get

\sum _{n\in \mathbb{Z}}(\mathrm{\nabla }W(n,u_k(n))-\mathrm{\nabla }W(n,u_0(n)),u_k(n)-u_0(n))\to 0\text{as }k\to +\mathrm{\infty }.

(3.10)

It follows from (2.17) that

\begin{array}{c}〈f^{\prime }(u_k)-f^{\prime }(u_0),u_k-u_0〉\hfill \\ ={\parallel u_k-u_0\parallel }^2-\sum _{n\in \mathbb{Z}}(\mathrm{\nabla }W(n,u_k(n))-\mathrm{\nabla }W(n,u_0(n)),u_k(n)-u_0(n)).\hfill \end{array}

(3.11)

Since 〈f^{\prime }(u_k)-f^{\prime }(u_0),u_k-u_0〉\to 0, it follows from (3.10) and (3.11) that u_k\to u_0 in E. Hence, f satisfies the (PS)-condition.

By Lemma 2.5, c={inf}_{E}f(u) is a critical value of f, that is, there exists a critical point u^{\ast }\in E such that f(u^{\ast })=c.

Finally, we show that u^{\ast }\ne 0. Let u_{\ast }(n_0)={(1,0,\dots ,0)}^{\mathrm{\top }}\in {\mathbb{R}}^{\mathcal{N}} and u_{\ast }(n)=0 for n\ne n_0. Then by (W1′), (W3′) and (2.16), we have

(3.12)

Since , it follows from (3.12) that for s>0 small enough. Hence , therefore u^{\ast } is a nontrivial critical point of f, and so u^{\ast }=u^{\ast }(n) is a nontrivial homoclinic solution of system (1.1). The proof is complete. □

Proof of Theorem 1.2 In view of Lemma 2.4 and the proof of Theorem 1.1, f\in C^1(E,\mathbb{R}) is bounded from below and satisfies the (PS)-condition. It is obvious that f is even and f(0)=0. In order to apply Lemma 2.6, we prove now that there is a set K\subset E such that K is homeomorphic to S^{m-1} by an odd map, and . Let

J=\{n_1,n_2,\dots ,n_m\},

where . Define

u_i(n)=\{\begin{array}{cc}{(1,0,\dots ,0)}^{\mathrm{\top }}\in {\mathbb{R}}^{\mathcal{N}},\hfill & n=n_i,\hfill \\ 0,\hfill & n\ne n_i;\hfill \end{array}i=1,2,\dots ,m

(3.13)

and

E_m=span\{u_1,u_2,\dots ,u_m\},K_m=\{u\in E_m:\parallel u\parallel =1\}.

(3.14)

For any u\in E_m, there exist {\lambda }_i\in \mathbb{R}, i=1,2,\dots ,m such that

u(n)=\sum _{i=1}^m{\lambda }_i{u}_i(n)\text{for }n\in \mathbb{Z}.

(3.15)

Then

{\parallel u\parallel }_{{\gamma }_4}={\left(\sum _{n\in \mathbb{Z}}{|u(n)|}^{{\gamma }_4}\right)}^{1/{\gamma }_4}={\left(\sum _{i=1}^m{|{\lambda }_i|}^{{\gamma }_4}{|u_i(n_i)|}^{{\gamma }_4}\right)}^{1/{\gamma }_4}

(3.16)

and

\begin{array}{rcl}{\parallel u\parallel }^2& =& \sum _{n\in \mathbb{Z}}[(p(n+1)\mathrm{△}u(n),\mathrm{△}u(n))+(L(n)u(n),u(n))]\\ =& \sum _{n\in \mathbb{Z}}(p(n+1)\sum _{i=1}^m{\lambda }_i\mathrm{△}{u}_i(n),\sum _{j=1}^m{\lambda }_j\mathrm{△}{u}_j(n))+\sum _{i=1}^m{\lambda }_i^2(L(n_i)u_i(n_i),u_i(n_i))\\ =& \sum _{i=1}^m\sum _{j=1}^m{\lambda }_i{\lambda }_j\sum _{n\in \mathbb{Z}}(p(n+1)\mathrm{△}{u}_i(n),\mathrm{△}{u}_j(n))+\sum _{i=1}^m{\lambda }_i^2(L(n_i)u_i(n_i),u_i(n_i))\\ =& F({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m),\end{array}

(3.17)

where F({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m) is a quadratic form. Since

F({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m)={\parallel \sum _{i=1}^m{\lambda }_i{u}_i\parallel }^2\ge 0,\mathrm{\forall }{({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m)}^{\mathrm{\top }}\in {\mathbb{R}}^m

and

F({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m)=0\iff \sum _{i=1}^m{\lambda }_i{u}_i(n)\equiv 0,\mathrm{\forall }n\in \mathbb{Z}\iff {\lambda }_1={\lambda }_2=\cdots ={\lambda }_m=0.

Therefore, F({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m) is a positive definite quadratic form. It follows that there exists an invertible matrix \mathcal{A}\in {\mathbb{R}}^{m\times m} such that

{({\mu }_1,{\mu }_2,\dots ,{\mu }_m)}^{\mathrm{\top }}=\mathcal{A}{({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m)}^{\mathrm{\top }},F({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m)=\sum _{i=1}^m{\mu }_i^2.

(3.18)

Since all the norms of a finite dimensional normed space are equivalent, there is a constant c^{\prime }>0 such that

c^{\prime }\parallel u\parallel \le {\parallel u\parallel }_{{\gamma }_4}\text{for }u\in E_m.

(3.19)

By (W1′), (W4′), (3.14), (3.16) and (3.19), we have

(3.20)

For , (3.20) implies that there exist \epsilon >0 and \sigma >0 such that

(3.21)

Let

K_m^{\sigma }=\{\sigma u:u\in K_m\},S^{m-1}=\{{({\mu }_1,{\mu }_2,\dots ,{\mu }_m)}^{\mathrm{\top }}\in {\mathbb{R}}^m:\sum _{i=1}^m{\mu }_i^2=1\}.

Then it follows from (3.17) that

K_m^{\sigma }=\{\sum _{i=1}^m{\lambda }_i{u}_i:F({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m)={\sigma }^2\}.

By (3.18), we define a map \psi :K_m^{\sigma }\to S^{m-1} as follows:

\psi (u)={\sigma }^{-1}{({\mu }_1,{\mu }_2,\dots ,{\mu }_m)}^{\mathrm{\top }},\mathrm{\forall }u\in K_m^{\sigma }.

It is easy to verify that \psi :K_m^{\sigma }\to S^{m-1} is an odd homeomorphic map. On the other hand, by (3.21), we have

and so . By Lemma 2.4, f has at least m distinct pairs of critical points, and so system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions. The proof is complete. □

4 Examples

In this section, we give three examples to illustrate our results.

Example 4.1 In system (1.1), let p(n) be an \mathcal{N}\times \mathcal{N} real symmetric positive definite matrix for all n\in \mathbb{Z}, L(n)=(1+{sin}^{2}n){|n|}^{2/3}{I}_{\mathcal{N}}, and let

W(n,x)=cosn{|x|}^{4/3}+sinn{|x|}^{3/2}.

Then L(n) satisfies (L _ν ) with \nu =4/3, and

\begin{array}{c}\mathrm{\nabla }W(n,x)=\frac{4cosn}{3}{|x|}^{-2/3}x+\frac{3sinn}{2}{|x|}^{-1/2}x,\hfill \\ |W(n,x)|\le {|x|}^{4/3}+{|x|}^{3/2},\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}}\hfill \end{array}

and

|\mathrm{\nabla }W(n,x)|\le \frac{8{|x|}^{1/3}+9{|x|}^{1/2}}{6},\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}}.

For any m\in \mathbb{N}, there exist m integers n_i\in \mathbb{Z}, i=1,2,\dots ,m such that

cos{n}_i>0,sin{n}_i>0,i=1,2,\dots ,m.

Let J=\{n_1,n_2,\dots ,n_m\} and

\eta =min\{cos{n}_1+sin{n}_1,cos{n}_2+sin{n}_2,\dots ,cos{n}_m+sin{n}_m\}.

Then

W(n,x)\ge \eta {|x|}^{3/2},\mathrm{\forall }(n,x)\in J\times {\mathbb{R}}^{\mathcal{N}},|x|\le 1.

These show that all conditions of Theorem 1.2 are satisfied, where

By Theorem 1.2, system (1.1) has at least m distinct pairs of nontrivial homoclinic solutions. Since m is arbitrary, it follows that system (1.1) has infinitely many distinct pairs of nontrivial homoclinic solutions.

Example 4.2 In system (1.1), let p(n) be an \mathcal{N}\times \mathcal{N} real symmetric positive definite matrix for all n\in \mathbb{Z}, L(n)=(1+{cos}^{2}n)|n|I_{\mathcal{N}}, and let

W(n,x)=sin(n+{|x|}^2)({|x|}^{3/2}-2{|x|}^{7/4}).

Then L(n) satisfies (L _ν ) with \nu =1, and

\begin{array}{c}\mathrm{\nabla }W(n,x)=sin(n+{|x|}^2)(\frac{3}{2}{|x|}^{-1/2}x-\frac{7}{4}{|x|}^{-1/4}x)\hfill \\ \phantom{\mathrm{\nabla }W(n,x)=}+2cos(n+{|x|}^2)({|x|}^{3/2}-2{|x|}^{7/4})x,\hfill \\ |W(n,x)|\le {|x|}^{3/2}+2{|x|}^{7/4},\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}},\hfill \\ |\mathrm{\nabla }W(n,x)|\le \frac{6{|x|}^{1/2}+7{|x|}^{3/4}+8{|x|}^{5/2}+16{|x|}^{11/4}}{4},\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}}.\hfill \end{array}

For any m\in \mathbb{N}, there exist m integers n_i\in \mathbb{Z}, i=1,2,\dots ,m such that

sin{n}_i>0,sin(n_i+1)>0,i=1,2,\dots ,m.

Let J=\{n_1,n_2,\dots ,n_m\} and

\eta =\frac{1}{2}min\{sin{n}_i,sin(n_i+1):i=1,2,\dots ,m\}.

Then

W(n,x)\ge \eta {|x|}^{3/2},\mathrm{\forall }(n,x)\in J\times {\mathbb{R}}^{\mathcal{N}},|x|\le \frac{1}{256}.

These show that all conditions of Theorem 1.2 are satisfied, where

By Theorem 1.2, system (1.1) has at least m distinct pairs of nontrivial homoclinic solutions. Since m is arbitrary, it follows that system (1.1) has infinitely many distinct pairs of nontrivial homoclinic solutions.

Example 4.3 In system (1.1), let p(n) be an \mathcal{N}\times \mathcal{N} real symmetric positive definite matrix for all n\in \mathbb{Z}, L(n)=ln(3+n^2){|n|}^{4/3}{I}_{\mathcal{N}}, and let

W(n,x)=sinnln(1+{|x|}^{3/2}).

Then L(n) satisfies (L _ν ) with \nu =2/3, and

\begin{array}{c}\mathrm{\nabla }W(n,x)=\frac{3sinn}{2(1+{|x|}^{3/2})}{|x|}^{-1/2}x,\hfill \\ |W(n,x)|\le {|x|}^{3/2},|\mathrm{\nabla }W(n,x)|\le \frac{3{|x|}^{1/2}}{2(1+{|x|}^{3/2})},\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}}.\hfill \end{array}

For any m\in \mathbb{N}, there exist m integers n_i\in \mathbb{Z}, i=1,2,\dots ,m such that

sin{n}_i>0,i=1,2,\dots ,m.

Let J=\{n_1,n_2,\dots ,n_m\} and

\eta =min\{\frac{sin{n}_i}{2}:i=1,2,\dots ,m\}.

Then

W(n,x)\ge \eta {|x|}^{3/2},\mathrm{\forall }(n,x)\in J\times {\mathbb{R}}^{\mathcal{N}},|x|\le 1.

These show that all conditions of Theorem 1.2 are satisfied, where

By Theorem 1.2, system (1.1) has at least m distinct pairs of nontrivial homoclinic solutions. Since m is arbitrary, it follows that system (1.1) has infinitely many distinct pairs of nontrivial homoclinic solutions.