## Boundary Value Problems

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# Ground state homoclinic orbits of damped vibration problems

Boundary Value Problems20142014:106

DOI: 10.1186/1687-2770-2014-106

Accepted: 24 April 2014

Published: 9 May 2014

## Abstract

In this paper, we consider a class of non-periodic damped vibration problems with superquadratic nonlinearities. We study the existence of nontrivial ground state homoclinic orbits for this class of damped vibration problems under some conditions weaker than those previously assumed. To the best of our knowledge, there has been no work focused on this case.

MSC: 49J40, 70H05.

### Keywords

non-periodic damped vibration problems ground state homoclinic orbits superquadratic nonlinearity

## 1 Introduction and main results

We shall study the existence of ground state homoclinic orbits for the following non-periodic damped vibration system:
(1.1)

where M is an antisymmetric constant matrix, is a symmetric matrix, and denotes its gradient with respect to the u variable. We say that a solution of (1.1) is homoclinic (to 0) if such that and as . If , then is called a nontrivial homoclinic solution.

If (zero matrix), then (1.1) reduces to the following second-order Hamiltonian system:
(1.2)

This is a classical equation which can describe many mechanic systems such as a pendulum. In the past decades, the existence and multiplicity of periodic solutions and homoclinic orbits for (1.2) have been studied by many authors via variational methods; see [118] and the references therein.

The periodic assumptions are very important in the study of homoclinic orbits for (1.2) since periodicity is used to control the lack of compactness due to the fact that (1.2) is set on all . However, non-periodic problems are quite different from the ones described in periodic cases. Rabinowitz and Tanaka [11] introduced a type of coercive condition on the matrix ,
(1.3)
and obtained the existence of a homoclinic orbit for non-periodic (1.2) under the Ambrosetti-Rabinowitz (AR) superquadratic condition:

where is a constant, denotes the standard inner product in and the associated norm is denoted by .

We should mention that in the case where , i.e., the damped vibration system (1.1), only a few authors have studied homoclinic orbits of (1.1); see [1923]. Zhu [23] considered the periodic case of (1.1) (i.e., and are T-periodic in t with ) and obtained the existence of nontrivial homoclinic solutions of (1.1). The authors [1922] considered the non-periodic case of (1.1): Zhang and Yuan [22] obtained the existence of at least one homoclinic orbit for (1.1) when H satisfies the subquadratic condition at infinity by using a standard minimizing argument; by a symmetric mountain pass theorem and a generalized mountain pass theorem, Wu and Zhang [21] obtained the existence and multiplicity of homoclinic orbits for (1.1) when H satisfies the local (AR) superquadratic growth condition:
(1.4)

where and are two constants. Notice that the authors [21, 22] all used condition (1.3). Recently, the author in [19, 20] obtained infinitely many homoclinic orbits for (1.1) when H satisfies the subquadratic[19] and asymptotically quadratic[20] condition at infinity by the following weaker conditions than (1.3):

(L1) There is a constant such that
(L2) There is a constant such that

which were firstly used in [15]. It is not hard to check that the matrix-valued function satisfies (L1) and (L2), but does not satisfy (1.3).

We define an operator by

Since M is an antisymmetric constant matrix, Γ is self-adjoint on . Let χ denote the self-adjoint extension of the operator . We are interested in the indefinite case:

(J1) .

To state our main result, we still need the following assumptions:

(H1) for some and , and .

(H2) , and .

(H3) For some and ,
(H4) as and there exists such that
(1.5)
(H5) For all and , there holds

Our main results read as follows.

Theorem 1.1If (L1)-(L2), (J1) and (H1)-(H5) hold, then (1.1) has at least one nontrivial homoclinic orbit.

Theorem 1.2Let be the collection of solutions of (1.1), then there is a solution that minimizes the energy functional
over , where the spaceEis defined in Section  2. In addition, if

uniformly int, then there is a nontrivial homoclinic orbit that minimizes the energy functional over, i.e., a ground state homoclinic orbit.

Remark 1.1 Although the authors [21] have studied (1.1) with superquadratic nonlinearities, our superquadratic condition (H4) is weaker than (1.4) in [21]. Moreover, we study the ground state homoclinic orbit of (1.1). To the best of our knowledge, there has been no result published concerning the ground state homoclinic orbit of (1.1).

Example 1.1
1. (1)

,

2. (2)

,

where , is continuous and . It is easy to check that the above two functions satisfy assumptions (H1)-(H5) if we take , where is the function in (H4)-(H5).

The rest of the present paper is organized as follows. In Section 2, we establish the variational framework associated with (1.1), and we also give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.

## 2 Preliminary lemmas

In the following, we use to denote the norm of for any . Let be a Hilbert space with the inner product and norm given respectively by
It is well known that W is continuously embedded in for . We define an operator by

Since M is an antisymmetric constant matrix, Γ is self-adjoint on W. Moreover, we denote by χ the self-adjoint extension of the operator with the domain .

Let , the domain of . We define respectively on E the inner product and the norm

where denotes the inner product in .

By a similar proof of Lemma 3.1 in [15], we can prove that if conditions (L1) and (L2) hold, then
(2.1)
Therefore, it is easy to prove that the spectrum has a sequence of eigenvalues (counted with their multiplicities)

and the corresponding system of eigenfunctions () forms an orthogonal basis in .

By (J1), we may let
Then one has the orthogonal decomposition
with respect to the inner product . Now, we introduce respectively on E the following new inner product and norm:
(2.2)
where with and . Clearly, the norms and are equivalent (see [4]), and the decomposition is also orthogonal with respect to both inner products and . Hence, by (J1), E with equivalent norms, besides, we have
(2.3)
and
(2.4)

where a and b are defined in (J1).

For problem (1.1), we consider the following functional:
Then I can be rewritten as
Let . In view of the assumptions of H, we know and the derivatives are given by

for any with and . By the discussion of [24], the (weak) solutions of system (1.1) are the critical points of the functional . Moreover, it is easy to verify that if is a solution of (1.1), then and as (see Lemma 3.1 in [25]).

The following abstract critical point theorem plays an important role in proving our main result. Let E be a Hilbert space with the norm and have an orthogonal decomposition , is a closed and separable subspace. There exists a norm satisfying for all and inducing a topology equivalent to the weak topology of N on a bounded subset of N. For with , , we define . Particularly, if is -bounded and , then weakly in N, strongly in , weakly in E (cf.[26]).

Let , with . Let and . For , let
with , . We define
For , define

where denotes various finite-dimensional subspaces of E, since .

The variant weak linking theorem is as follows.

Lemma 2.1 ([26])

The family of-functionalshas the form
where. Assume that
1. (a)

, , ;

2. (b)

as;

3. (c)

is-upper semicontinuous, is weakly sequentially continuous onE. Moreover, maps bounded sets to bounded sets;

4. (d)

, .

Then, for almost all, there exists a sequencesuch that

where.

In order to apply Lemma 2.1, we shall prove a few lemmas. We pick such that . For , we consider
It is easy to see that satisfies condition (a) in Lemma 2.1. To see (c), if and , then and in E, a.e. on , going to a subsequence if necessary. It follows from the weak lower semicontinuity of the norm, Fatou’s lemma and the fact for all and by (1.5) in (H4) that

Thus we get . It implies that is -upper semicontinuous. is weakly sequentially continuous on E due to [27].

Lemma 2.2Under assumptions of Theorem  1.1, then
Proof By the definition of and (H4), we have

which is due to . □

Therefore, Lemma 2.2 implies that condition (b) holds. To continue the discussion, we still need to verify condition (d), that is, the following two lemmas.

Lemma 2.3Under assumptions of Theorem  1.1, there are two positive constantssuch that
Proof By (H1), (H3), (2.4) and the Sobolev embedding theorem, for all ,

where C is a positive constant. It implies the conclusion if we take sufficiently small. □

Lemma 2.4Under assumptions of Theorem  1.1, then there is ansuch that

where.

Proof Suppose by contradiction that there exist , and such that . If , then by (H2) and (2.3), we have
Therefore, and . Let , then
It follows from and the definition of I that
(2.5)

There are renamed subsequences such that , , and there is a renamed subsequence such that in E and a.e. on .

We claim that
(2.6)
Case 1. If . Let be the subset of where , then for all we have . It follows from (H4) and that
Case 2. If , then by (H4) and , we have
Therefore, Cases 1 and 2 imply that (2.6) holds. Therefore, by (2.5), (2.6) and the facts , , we have
that is, . Thus, . It follows from (H4) that

which contradicts (2.5). The proof is finished. □

Therefore, Lemmas 2.3 and 2.4 imply that condition (d) of Lemma 2.1 holds. Applying Lemma 2.1, we soon obtain the following fact.

Lemma 2.5Under assumptions of Theorem  1.1, for almost all, there exists a sequencesuch that

where the definition ofis given in Lemma  2.1.

Lemma 2.6Under assumptions of Theorem  1.1, for almost all, there exists asuch that
Proof Let be the sequence obtained in Lemma 2.5. Since is bounded, we can assume in E and a.e. on . By (H1), (H3), (2.1) and Theorem A.4 in [27], we have
(2.7)
and
(2.8)
By Lemma 2.5 and the fact that is weakly sequentially continuous, we have
That is, . By Lemma 2.5, we have
It follows from (2.7), (2.8) and the fact that

The proof is finished. □

Applying Lemma 2.6, we soon obtain the following fact.

Lemma 2.7Under assumptions of Theorem  1.1, for every, there are sequencesandwithsuch that
Lemma 2.8Under assumptions of Theorem  1.1, then

where, , and the constantdoes not depend onu, wr.

Proof This follows from (H5) if we take and . □

Lemma 2.9The sequences given in Lemma  2.7 are bounded.

Proof Write , where . Suppose that

Let , then , , and . Thus in E and a.e. on , after passing to a subsequence.

Case 1. If . Let be the subset of where . Then and on . It follows from (H4) and that
which together with Lemmas 2.3 and 2.7 and in for all (by (2.1)) implies that

Case 2. If . We claim that there is a constant C independent of and such that
(2.9)
Since
it follows from the definition of I that
(2.10)
Take in (2.10), then it follows from Lemma 2.8 that

Thus (2.9) holds.

Let be a fixed constant and take
Therefore, (2.9) implies that
It follows from and Lemma 2.7 that
(2.11)
Note that Lemmas 2.3 and 2.7 and (H4) imply that
It follows from the fact as due to that
(2.12)
for all sufficiently large n. We take , by (2.12) and , we have
(2.13)
for all sufficiently large n. By (H1) and (H3), we have
(2.14)
For all sufficiently large n, by (2.13) and (2.14), it follows from and in for all (by (2.1)) that

This implies that as , contrary to (2.11).

Therefore, are bounded. The proof is finished. □

## 3 Proofs of the main results

Proof of Theorem 1.1 From Lemma 2.7, there are sequences and such that and . By Lemma 2.9, we know that is bounded in E. Thus we can assume in E, a.e. on . Therefore,
Hence, in the limit,
Thus . Note that
(3.1)
Similar to (2.7) and (2.8), we know
It follows from , (3.1) and Lemma 2.3 that

Therefore, . □

Proof of Theorem 1.2 By Theorem 1.1, , where is the collection of solutions of (1.1). Let
If u is a solution of (1.1), then by Lemma 2.8 (take ),
Thus . Let be a sequence in such that
(3.2)
By Lemma 2.9, the sequence is bounded in E. Therefore, in E, a.e. on and in for all (by (2.1)), after passing to a subsequence. Therefore,
Hence, in the limit,
Thus . Similar to (2.7) and (2.8), we have
It follows from and (3.2) that
Now suppose that
It follows from (H1) that for any , there is a constant such that
(3.3)
Let
where . Let be a sequence in such that
(3.4)
Note that
which together with (3.3), Hölder’s inequality and the Sobolev embedding theorem implies
(3.5)
Similarly, we have
(3.6)
From (3.5) and (3.6), we get

which means for some constant . Since in , we know . As before, as . □

## Declarations

### Acknowledgements

The authors thank the referees and the editors for their helpful comments and suggestions. Research was supported by the Tianyuan Fund for Mathematics of NSFC (Grant No. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant No. 13A110015).

## Authors’ Affiliations

(1)
School of Mathematics and Statistics, Anyang Normal University

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