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Ground state homoclinic orbits of superquadratic damped vibration systems
Advances in Difference Equations volume 2014, Article number: 230 (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
We shall study the existence of ground state homoclinic orbits for the following 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 [1–17] and the references therein.
The periodicity assumption is 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 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 the case where , i.e., the damped vibration system (1.1), only a few authors have studied homoclinic orbits of (1.1); see [18–22]. 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 [18–21] 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 superquadratic growth condition:
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):
(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 satisfying (L1) and (L2), but not satisfying (1.3).
We define an operator by
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 , .
(H3) if , and for any we have
(H4) There are constants and such that
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).
Example 1.1 Let
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. ).
Let , with . Let and . For , let
with , . We define
For , we define
where denotes various finite-dimensional subspaces of , since .
We shall use the following variant weak linking theorem to prove our result.
Theorem A ()
The family of -functional has the form
, , ;
or as ;
is -upper semicontinuous, is weakly sequentially continuous on W. Moreover, maps bounded sets to bounded sets;
Then for almost all , there exists a sequence such that
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.
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 E is continuously embedded in for . We define an operator by
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 .
Let , the domain of . We define, respectively, on W the inner product and the norm
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 .
By Lemma 2.1, 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 W 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 .
For problem (1.1), we consider the following functional:
Then Φ 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 ).
In order to apply Theorem A, we consider
It is easy to see that satisfies conditions (a), (b) in Theorem A. To see (c), if , then and in W, going to a subsequence if necessary, a.e. on ℝ. By Fatou’s lemma and the weak lower semicontinuity of the norm, we have
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.
Lemma 2.2 Under the assumptions of Theorem 1.1, we have the following facts:
There exists independent of such that , where
For fixed with and any , there is such that , where .
Proof (i) Under assumptions (H1) and (H4), we know for any there exists such that
where with . Hence, for any ,
which implies the conclusion.
Suppose by contradiction that there exist such that for all n and as . Let , then(2.5)
It follows from (see (H2)) that
therefore, and .
Thus after passing to a subsequence, and a.e. on ℝ. Hence, and, since if , it follows from (H2) and Fatou’s lemma that
contrary to (2.5). The proof is finished. □
Applying Theorem A, we soon obtain the following facts.
Lemma 2.3 Under the assumptions of Theorem 1.1, for almost all , there exists a sequence such that
Lemma 2.4 Under the assumptions of Theorem 1.1, for almost all , there exists a such that
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.
We claim that . If not, then Lemma 2.1 implies in for all . It follows from the definition of Φ, Hölder’s inequality, and (2.3) that
It follows from (2.2) and Lemma 2.3 that
which contradicts with the fact that . Hence, , and thus . Note that is weakly sequentially continuous on W, thus
By (H3), Fatou’s lemma, and Lemma 2.3, we have
Thus we get . □
Lemma 2.5 Under the assumptions of Theorem 1.1, there exists and such that
Moreover, is bounded.
Proof The existence of such that
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.
Recall that . Thus for any , we have
Let in (2.7), respectively. Then we have
Since , we have
For , let
By (H1) and (H3), we have for all . By (H3) and (H4), for ,
it follows from (H2) and the definition of that
For , let
By (H3), we have and
Since and , there exists a constant such that for all n
from which we have
Invoking (H4), set and . Since one sees . Fix arbitrarily . By (2.10) and the fact as , we have
as uniformly in n, it follows from , Hölder’s inequality, and Sobolev’s embedding theorem that
as uniformly in n. By (2.10) and , for any fixed ,
Let . Sobolev’s embedding theorem implies and . It follows from the fact that there is such that for all (see (H1)) that
for all n. By (H4), (2.9), and (2.11), we can take large so that
for all n. Note that there is independent of n such that for . By (2.12) there is such that
for all . Therefore, the combination of (2.13)-(2.15) implies that for , we have
which contradicts with (2.8). Thus is bounded. □
Lemma 2.6 If is the sequence obtained in Lemma 2.5, then it is also a sequence for Φ satisfying
Proof Note that is bounded. From
and noting that
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.
By (2.3), , Hölder’s inequality, and Sobolev’s embedding theorem,
Similarly, we have
From (3.1) and (3.2), we get
which means for some constant c, it follows from in that . The facts that is weakly sequentially continuous on W and in W imply .
Let be the critical set of Φ and
For any critical point u of Φ, assumption (H3) implies that
Therefore . We prove that and there is such that . Let be such that . Then the proof in Lemma 2.5 shows that is bounded, and by the concentration compactness principle discussion above we know . Thus
where the first inequality is due to (H3) and Fatou’s lemma. So and because . □
Ambrosetti A, Coti Zelati V: Multiple homoclinic orbits for a class of conservative systems. Rend. Semin. Mat. Univ. Padova 1993, 89: 177-194.
Chen G, Ma S: Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum. J. Math. Anal. Appl. 2011, 379: 842-851. 10.1016/j.jmaa.2011.02.013
Ding Y: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 1995, 25: 1095-1113. 10.1016/0362-546X(94)00229-B
Izydorek M, Janczewska J: Homoclinic solutions for a class of second order Hamiltonian systems. J. Differ. Equ. 2005, 219: 375-389. 10.1016/j.jde.2005.06.029
Kim Y: Existence of periodic solutions for planar Hamiltonian systems at resonance. J. Korean Math. Soc. 2011, 48: 1143-1152. 10.4134/JKMS.2011.48.6.1143
Mawhin J, Willem M Applied Mathematical Sciences 74. In Critical Point Theory and Hamiltonian Systems. Springer, New York; 1989.
Omana W, Willem M: Homoclinic orbits for a class of Hamiltonian systems. Differ. Integral Equ. 1992, 5: 1115-1120.
Paturel E: Multiple homoclinic orbits for a class of Hamiltonian systems. Calc. Var. Partial Differ. Equ. 2001, 12: 117-143. 10.1007/PL00009909
Rabinowitz PH: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb., Sect. A 1990, 114: 33-38. 10.1017/S0308210500024240
Rabinowitz PH, Tanaka K: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 1991, 206: 473-499. 10.1007/BF02571356
Séré E: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 1992, 209: 133-160.
Sun J, Chen H, Nieto JJ: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 2011, 373: 20-29. 10.1016/j.jmaa.2010.06.038
Tang X, Xiao L: Homoclinic solutions for non-autonomous second-order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl. 2009, 351: 586-594. 10.1016/j.jmaa.2008.10.038
Wan L, Tang C: Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete Contin. Dyn. Syst., Ser. B 2011, 15: 255-271.
Xiao J, Nieto JJ: Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J. Franklin Inst. 2011, 348: 369-377. 10.1016/j.jfranklin.2010.12.003
Zhang P, Tang C: Infinitely many periodic solutions for nonautonomous sublinear second-order Hamiltonian systems. Abstr. Appl. Anal. 2010., 2010: Article ID 620438 10.1155/2010/620438
Zhang Q, Liu C: Infinitely many homoclinic solutions for second order Hamiltonian systems. Nonlinear Anal. 2010, 72: 894-903. 10.1016/j.na.2009.07.021
Chen G: Non-periodic damped vibration systems with sublinear terms at infinity: infinitely many homoclinic orbits. Nonlinear Anal. 2013, 92: 168-176.
Chen G: Non-periodic damped vibration systems with asymptotically quadratic terms at infinity: infinitely many homoclinic orbits. Abstr. Appl. Anal. 2013., 2013: Article ID 937128
Wu X, Zhang W: Existence and multiplicity of homoclinic solutions for a class of damped vibration problems. Nonlinear Anal. 2011, 74: 4392-4398. 10.1016/j.na.2011.03.059
Zhang Z, Yuan R: Homoclinic solutions for some second-order nonautonomous systems. Nonlinear Anal. 2009, 71: 5790-5798. 10.1016/j.na.2009.05.003
Zhu W: Existence of homoclinic solutions for a class of second order systems. Nonlinear Anal. 2012, 75: 2455-2463. 10.1016/j.na.2011.10.043
Schechter M, Zou W: Weak linking theorems and Schrödinger equations with critical Sobolev exponent. ESAIM Control Optim. Calc. Var. 2003, 9: 601-619.
Costa DG, Magalhães CA: A variational approach to subquadratic perturbations of elliptic systems. J. Differ. Equ. 1994, 111: 103-122. 10.1006/jdeq.1994.1077
Wang J, Xu J, Zhang F: Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Commun. Pure Appl. Anal. 2011, 10: 269-286.
Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.
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).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Chen, G., Zhao, X. Ground state homoclinic orbits of superquadratic damped vibration systems. Adv Differ Equ 2014, 230 (2014). https://doi.org/10.1186/1687-1847-2014-230
- damped vibration systems
- ground state homoclinic orbits
- variant weak linking theorem