- Open Access
Ground state homoclinic orbits of superquadratic damped vibration systems
© Chen and Zhao; licensee Springer 2014
Received: 10 June 2014
Accepted: 7 August 2014
Published: 20 August 2014
In the case where nonlinearities are superquadratic at infinity, we study the existence of ground state homoclinic orbits for damped vibration systems without periodic conditions by using variational methods. Here the (local) Ambrosetti-Rabinowitz superquadratic condition is replaced by a general superquadratic condition.
1 Introduction and main result
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 [1–17] 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 [20, 21] all used the condition (1.3). Recently, Chen [18, 19] 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 satisfying (L1) and (L2), but not satisfying (1.3).
Since M is an antisymmetric constant matrix, J is self-adjoint on . Let χ denote the self-adjoint extension of the operator . We are interested in the indefinite case:
Let . We assume the following.
(H1) and as uniformly in t.
(H2) as uniformly in t, and , .
Now, our main result reads as follows.
Theorem 1.1 If (J1), (L1)-(L2), and (H1)-(H4) hold, then (1.1) has a ground state homoclinic orbit.
Remark 1.1 Although the authors  have studied the superquadratic case of (1.1), it is not hard to check that our superquadratic condition (H2) is weaker than the condition (1.4) (see Example 1.1). Moreover, we obtain the existence of ground state homoclinic orbits of (1.1), i.e., nontrivial homoclinic orbits with least energy of the action functional of (1.1).
where , and is continuous. It is not hard to check that H satisfies (H1)-(H4) but does not satisfy (1.4).
The following abstract critical point theorem plays an important role in proving our main result. Let W be a Hilbert space with norm and have an orthogonal decomposition , is a closed and separable subspace. There exists a norm that satisfies for all and induces a topology equivalent to the weak topology of N on bounded subset of N. For with , , we define . Particularly, if is -bounded and , then weakly in N, strongly in , weakly in W (cf. ).
where denotes various finite-dimensional subspaces of , since .
We shall use the following variant weak linking theorem to prove our result.
Theorem A ()
, , ;
or as ;
is -upper semicontinuous, is weakly sequentially continuous on W. Moreover, maps bounded sets to bounded sets;
The rest of the present paper is organized as follows. In Section 2, we give some preliminary lemmas, which are useful in the proof of our main result. In Section 3, we give the detailed proof of our main result.
In this section, we firstly give the variational frameworks of our problem and some related preliminary lemmas, and then give the detailed proof of the main result.
Since M is an antisymmetric constant matrix, J is self-adjoint on E. Moreover, we denote by χ the self-adjoint extension of the operator with the domain .
where denotes the inner product in .
By a similar proof of Lemma 3.1 in , we can prove the following lemma.
Lemma 2.1 If conditions (L1) and (L2) hold, then W is compactly embedded into for all .
and the corresponding system of eigenfunctions () forms an orthogonal basis in .
where with and . Clearly, the norms and are equivalent (see ), and the decomposition is also orthogonal with respect to both inner products and .
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 ).
which means that is -upper semicontinuous. is weakly sequentially continuous on W is due to . To continue the discussion, we still need to verify condition (d) in Theorem A.
- (i)There exists independent of such that , where
For fixed with and any , there is such that , where .
- (ii)Suppose by contradiction that there exist such that for all n and as . Let , then(2.5)
therefore, and .
contrary to (2.5). The proof is finished. □
Applying Theorem A, we soon obtain the following facts.
Proof Let be the sequence obtained in Lemma 2.3, write with . Since is bounded, is also bounded, then and in W, after passing to a subsequence.
Thus we get . □
Moreover, is bounded.
is the direct consequence of Lemma 2.4. To prove the boundedness of , arguing by contradiction, suppose that . Let . Then , in W and a.e. in ℝ, after passing to a subsequence.
which contradicts with (2.8). Thus is bounded. □
uniformly in , we obtain the conclusion. □
3 Proof of main result
We are now in a position to prove our main result.
Proof of Theorem 1.1 Note that Lemma 2.5 implies is bounded, thus in W, and in for all by Lemma 2.1, after passing to a subsequence.
which means for some constant c, it follows from in that . The facts that is weakly sequentially continuous on W and in W imply .
where the first inequality is due to (H3) and Fatou’s lemma. So and because . □
Research 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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