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.
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:
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):
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).
To state our main result, we still need the following assumptions:
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
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).
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
By a similar proof of Lemma 3.1 in , we can prove that if conditions (L1) and (L2) hold, then
By (J1), we may let
Then one has the orthogonal decomposition
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
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.).
The variant weak linking theorem is as follows.
Lemma 2.1 ()
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
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.
We claim 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.
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
The proof is finished. □
Applying Lemma 2.6, we soon obtain the following fact.
Lemma 2.8Under assumptions of Theorem 1.1, then
Lemma 2.9The sequences given in Lemma 2.7 are bounded.
It is a contradiction.
it follows from the definition of I that
Thus (2.9) holds.
Therefore, (2.9) implies that
Note that Lemmas 2.3 and 2.7 and (H4) imply that
for all sufficiently large n. By (H1) and (H3), we have
3 Proofs of the main results
Hence, in the limit,
Similar to (2.7) and (2.8), we know
Hence, in the limit,
Now suppose 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
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).
Chen, G, Ma, S: Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum. J. Math. Anal. Appl.. 379, 842–851 (2011). Publisher Full Text
Chen, G, Ma, S: Ground state periodic solutions of second order Hamiltonian systems without spectrum 0. Isr. J. Math.. 198, 111–127 (2013). Publisher Full Text
Ding, Y: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal.. 25, 1095–1113 (1995). Publisher Full Text
Izydorek, M, Janczewska, J: Homoclinic solutions for a class of second order Hamiltonian systems. J. Differ. Equ.. 219, 375–389 (2005). Publisher Full Text
Kim, Y: Existence of periodic solutions for planar Hamiltonian systems at resonance. J. Korean Math. Soc.. 48, 1143–1152 (2011). Publisher Full Text
Paturel, E: Multiple homoclinic orbits for a class of Hamiltonian systems. Calc. Var. Partial Differ. Equ.. 12, 117–143 (2001). Publisher Full Text
Rabinowitz, PH: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb., Sect. A. 114, 33–38 (1990). Publisher Full Text
Rabinowitz, PH, Tanaka, K: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z.. 206, 473–499 (1991). Publisher Full Text
Sun, J, Chen, H, Nieto, JJ: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl.. 373, 20–29 (2011). Publisher Full Text
Tang, X, Xiao, L: Homoclinic solutions for non-autonomous second-order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl.. 351, 586–594 (2009). Publisher Full Text
Xiao, J, Nieto, JJ: Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J. Franklin Inst.. 348, 369–377 (2011). Publisher Full Text
Article ID 620438Publisher Full Text
Zhang, Q, Liu, C: Infinitely many homoclinic solutions for second order Hamiltonian systems. Nonlinear Anal.. 72, 894–903 (2010). Publisher Full Text
Wu, X, Zhang, W: Existence and multiplicity of homoclinic solutions for a class of damped vibration problems. Nonlinear Anal.. 74, 4392–4398 (2011). Publisher Full Text
Zhang, Z, Yuan, R: Homoclinic solutions for some second-order nonautonomous systems. Nonlinear Anal.. 71, 5790–5798 (2009). Publisher Full Text
Zhu, W: Existence of homoclinic solutions for a class of second order systems. Nonlinear Anal.. 75, 2455–2463 (2012). Publisher Full Text
Costa, DG, Magalhães, CA: A variational approach to subquadratic perturbations of elliptic systems. J. Differ. Equ.. 111, 103–122 (1994). Publisher Full Text