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:
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:
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 [1-18] 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 ℝ. However, non-periodic problems are quite different from the ones described in periodic cases. Rabinowitz and Tanaka  introduced a type of coercive condition on the matrix ,
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 [19-23]. Zhu  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 [19-22] considered the non-periodic case of (1.1): Zhang and Yuan  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  obtained the existence and multiplicity of homoclinic orbits for (1.1) when H satisfies the local (AR) superquadratic growth condition:
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 and asymptotically quadratic 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 . 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:
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
(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 ℳ be the collection of solutions of (1.1), then there is a solution that minimizes the energy functional
over ℳ, 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  have studied (1.1) with superquadratic nonlinearities, our superquadratic condition (H4) is weaker than (1.4) in . 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).
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 , we can prove that if conditions (L1) and (L2) hold, then
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:
where with and . Clearly, the norms and are equivalent (see ), and the decomposition is also orthogonal with respect to both inner products and . Hence, by (J1), E with equivalent norms, besides, we have
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 , 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 ).
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.).
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 ()
The family of -functionals has the form
where . Assume that
(a) , , ;
(b) as ;
(c) is -upper semicontinuous, is weakly sequentially continuous onE. Moreover, maps bounded sets to bounded sets;
(d) , .
Then, for almost all , there exists a sequence such that
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 ℝ, 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 .
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 constants such 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 an such that
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
There are renamed subsequences such that , , and there is a renamed subsequence such that in E and a.e. on ℝ.
We claim that
Case 1. If . Let be the subset of ℝ 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 sequence such that
where the definition of is given in Lemma 2.1.
Lemma 2.6Under assumptions of Theorem 1.1, for almost all , there exists a such that
Proof Let be the sequence obtained in Lemma 2.5. Since is bounded, we can assume in E and a.e. on ℝ. By (H1), (H3), (2.1) and Theorem A.4 in , we have
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 sequences and with such that
Lemma 2.8Under assumptions of Theorem 1.1, then
where , , and the constant does not depend onu, w, r.
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 ℝ, after passing to a subsequence.
Case 1. If . Let be the subset of ℝ 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
It is a contradiction.
Case 2. If . We claim that there is a constant C independent of and such that
it follows from the definition of I that
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
Note that Lemmas 2.3 and 2.7 and (H4) imply that
It follows from the fact as due to that
for all sufficiently large n. We take , by (2.12) and , we have
for all sufficiently large n. By (H1) and (H3), we have
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 ℝ. Therefore,
Hence, in the limit,
Thus . Note that
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 ℳ 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 ℳ such that
By Lemma 2.9, the sequence is bounded in E. Therefore, in E, a.e. on ℝ 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
where . Let be a sequence in such that
which together with (3.3), Hölder’s inequality and the Sobolev embedding theorem implies
Similarly, we have
From (3.5) and (3.6), we get
which means for some constant . Since in , we know . As before, as . □
The authors declare that they have no competing interests.
The main idea of this paper was proposed by G-WC and G-WC prepared the manuscript initially and JW performed a part of steps of the proofs in this research. All authors read and approved the final manuscript.
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).
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