Ground state homoclinic orbits of damped vibration problems
© Chen and Wang; licensee Springer. 2014
Received: 6 January 2014
Accepted: 24 April 2014
Published: 9 May 2014
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.
Keywordsnon-periodic damped vibration problems ground state homoclinic orbits superquadratic nonlinearity
1 Introduction and main results
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.
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.
where is a constant, denotes the standard inner product in and the associated norm is denoted by .
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):
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).
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 .
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.
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
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 .
where denotes the inner product in .
and the corresponding system of eigenfunctions () forms an orthogonal basis in .
where a and b are defined in (J1).
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.).
where denotes various finite-dimensional subspaces of E, since .
The variant weak linking theorem is as follows.
Lemma 2.1 ()
, , ;
is-upper semicontinuous, is weakly sequentially continuous onE. Moreover, maps bounded sets to bounded sets;
Thus we get . It implies that is -upper semicontinuous. is weakly sequentially continuous on E due to .
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.
where C is a positive constant. It implies the conclusion if we take sufficiently small. □
There are renamed subsequences such that , , and there is a renamed subsequence such that in E and a.e. on ℝ.
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.
where the definition ofis given in Lemma 2.1.
The proof is finished. □
Applying Lemma 2.6, we soon obtain the following fact.
where, , and the constantdoes 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.
Let , then , , and . Thus in E and a.e. on ℝ, after passing to a subsequence.
It is a contradiction.
Thus (2.9) holds.
This implies that as , contrary to (2.11).
Therefore, are bounded. The proof is finished. □
3 Proofs of the main results
Therefore, . □
which means for some constant . Since in , we know . As before, as . □
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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