Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits
We study a class of nonperiodic damped vibration systems with asymptotically quadratic terms at infinity. We obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by Zou. To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for this class of non-periodic damped vibration systems with asymptotically quadratic terms at infinity.
1. Introduction and Main Results
In the end of 19th century, Poincaré recognized the importance of homoclinic orbits for dynamical systems. Since then the existence and multiplicity of homoclinic solutions have become one of the most important problems in the research of dynamical systems. In this paper, we consider the following nonperiodic damped vibration system (NDVS): where is an antisymmetric constant matrix,is a symmetric matrix, anddenotes its gradient with respect to thevariable. We say that a solutionof (1) is homoclinic (to 0) ifsuch that If, thenis called a nontrivial homoclinic solution.
If(zero matrix), then (1) reduces to the following second-order Hamiltonian system: which is a classical equation which can describe many mechanical systems, such as a pendulum. In the past decades, the existence and multiplicity of periodic solutions and homoclinic orbits for (3) have been studied by many authors via variational methods; see [1–17] and the references therein. The periodic assumptions are very important in the study of homoclinic orbits for (3) since periodicity is used to control the lack of compactness due to the fact that (3) is set on all.
Nonperiodic problems are quite different from the ones described in periodic cases. Rabinowitz and Tanaka  introduced a type of coercivity condition on the matrix: and obtained the existence of homoclinic orbit for nonperiodic (3) under the usual Ambrosetti-Rabinowitz superquadratic condition: whereis a constant,denotes the standard inner product in, and the associated norm is denoted by.
As usual, we say that satisfies the subquadratic (or superquadratic) growth condition at infinity if If , that is, the damped vibration system (1), there are only a few authors who have studied homoclinic orbits of the NDVS (1), see [18–23]. Zhu  considered the periodic case of (1) (i.e.,andare-periodic inwith) and obtained the existence of nontrivial homoclinic solutions of (1). The authors [19–23] considered the nonperiodic case of (1): Zhang and Yuan  obtained the existence of at least one homoclinic orbit for (1) whensatisfies 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) whensatisfies the localsuperquadratic growth condition: whereandare two constants. We should notice that the matrixin (1) is required to satisfy condition (4) in the Previously mentioned two papers [19, 20]. Later, Sun et al.  obtained the existence of at least one homoclinic orbit for (1) whensatisfies the superquadratic condition at infinity by using the following conditions which are weaker than condition (4).There exists a constantsuch that There exists a constantsuch that Recently, by using conditionsand, Chen [22, 23] obtained infinitely many nontrivial homoclinic orbits of (1) whensatisfies the subquadratic  (or superquadratic ) growth condition at infinity. In fact, conditions andare first used in . As mentioned in , there are some matrix-valued functionssatisfyingand but not satisfying (4). For example, That is, conditionsandare weaker than condition (4).
Remark 1. To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for the NDVS (1) whensatisfies the asymptotically quadratic condition at infinity (see the following condition).
Let. We assume the following. There are constantsandsuch that for all. uniformly in, where. asand
We should mention that the coercive-type assumption (see) of the functionwas first observed and used by Costa and Magalhães .
Now, our main result reads as follows.
Theorem 2. If,, –, and are even inhold, then (1) possesses infinitely many nontrivial homoclinic orbits.
Example 3. Let whereis defined in. It is not hard to check that it satisfies conditions– with in.
The rest of our paper is organized as follows. In Section 2, we establish the variational framework associated with (1) and give some preliminary lemmas, which are useful in the proof of our result, and then we give the detailed proof of our main result.
2. Variational Frameworks and the Proof of Our Main Result
In this section, we always assume that, , –, andare even inhold.
In the following, we will useto denote the norm offor any. Let be a Hilbert space with the inner product and the norm given, respectively, by It is well known thatis continuously embedded infor. We define an operator by Sinceis an antisymmetricconstant matrix,is self-adjoint on. Moreover, we denote bythe self-adjoint extension of the operatorwith the domain.
Let, the domain of. We define, respectively, onthe inner product and the norm wheredenotes the inner product in.
Lemma 4 (, Lemma 4). If conditionsandhold, thenis compactly embedded into for all .
By Lemma 4, it is easy to prove that the spectrumhas a sequence of eigenvalues (counted with their multiplicities) and the corresponding system of eigenfunctions forms an orthogonal basis in . Let Then, one has the orthogonal decomposition with respect to the inner product.
Now, we introduce, respectively, onthe following new inner product and norm: wherewith, and. Clearly, the two norms and are equivalent (see ), and the decompositionis also orthogonal with respect to both inner productsand.
For problem (1), we consider the following functional: Then,can be rewritten as Let. By the assumptions of, we know that and the derivatives are given by for anywithand. By the discussion of , the (weak) solutions of system (1) are the critical points of thefunctional. Moreover, it is easy to verify that ifis a solution of (1), then and as (see Lemma 3.1 in ).
Letbe a Banach space with the norm andwithfor any. Set Consider the following-functionaldefined by
To continue the discussion, we give the following variant fountain theorem.
Lemma 5 (see ). Assume that the functionaldefined previously satisfies maps bounded sets to bounded sets uniformly for, and for allandason any finite-dimensional subspace of ; there existsuch that Then, there existandsuch that Particularly, ifhas a convergent subsequence for every, thenhas infinitely many nontrivial critical pointssatisfyingas.
For, let(the sequenceis defined in Section 2 just below Lemma 4); thenandcan be defined as before. In order to apply the previously mentioned variant fountain theorem to prove our main result, we define the functionals , , andonby for all and . Obviously, for all .
Next, we will prove that conditionsandof Lemma 5 hold, that is, the following two lemmas.
Lemma 6. for allandason any finite-dimensional subspace of .
Proof. Obviously, conditionand the definition ofimply thatfor all. We claim that for any finite-dimensional subspace, there exists a constantsuch that
wheredenotes the Lebesgue measure in. In fact, the detailed proof of (30) has been given by Chen (Lemma 2.3 in ).
For thegiven in (30), let Then, by (30), By, there exist constantssuch that The definition ofimplies that for anywith there holds Combining, (32)–(34), and the definition of, for anywith, we have It implies that ason any finite-dimensional subspace. The proof is finished.
Lemma 7. There exist a positive integerand two sequencesassuch that whereandfor all.
Proof. (a) First, we show that (36) holds. Note thatfor all, whereis the integer defined in (17) just below Lemma 4. By Lemma 4, there is a constantsuch that for any. It follows that for anywith there holds
whereis the constant in. It follows fromand the definition ofthat for anyandwith there holds
by Lemma 4 and the Rellich embedding theorem (see ). Consequently, (40) and (41) imply that
for anyandwith . For any, let
Then, by (42), we have
Evidently, (45) implies that there exists a positive integersuch that
(43) together with (44) and (46) implies that
That is, (36) holds.
(b) Second, we show that (37) holds. By (43), for anyandwith , we have Observing thatby, thus which together with (42) and (45) implies that That is, (37) holds.
(c) Last, we show that (38) holds. For anyandwith (is the constant above (39)), similar to (39), we have Therefore, by (51) and, for anyandwith , we have where the last inequality follows by the equivalence of the two normsandon finite dimensional space, andis a constant depending on. For any, if we choose Then, by (52), direct computation shows that That is, (38) holds.
Therefore, the proof is finished by (a), (b), and (c).
Proof of Theorem 2. By the assumptions ofand the definition of, we easily get thatmaps bounded sets to bounded sets uniformly for. Note that, so we have for all. Thus, the conditionof Lemma 5 holds. Lemma 6 shows that the conditionof Lemma 5 holds. Lemma 7 implies that the conditionof Lemma 5 holds for all, whereis given in Lemma 7. Therefore, by Lemma 5, for each, there exist , such that
Next, we only need to prove the following two claims to complete the proof of Theorem 2.
Claim 1. is bounded in.
Proof of Claim 1. By (55), we have for some constant . It follows from the definitions ofandthat Sinceimpliesas, it follows from (57) that for some constant. Note that; it follows fromthat there is a constantsuch that Thus, by (57)–(59),, Hölder’s inequality, and Lemma 4, for some positive constantand. It implies that. On the other hand, and imply that that is, It follows fromthatis bounded in. Therefore, Claim 1 is true.
Claim 2. has a strongly convergent subsequence in.
Proof of Claim 2. Note that. By Claim 1, without loss of generality, we may assume that
for some. By virtue of the Riesz Representation Theorem, andcan be viewed asand, respectively, whereandare the dual spaces ofand, respectively. Note that
whereis the orthogonal projection for all; that is,
By the assumptions ofand the standard argument (see [29, 30]), we knowis compact. Therefore,is also compact. Due to the compactness ofand (63), the right-hand side of (65) converges strongly inand hencein. Combining this with (63), we have
Therefore, Claim 2 is true.
Now, from the last assertion of Lemma 5, we know thathas infinitely many nontrivial critical points. Therefore, (1) possesses infinitely many nontrivial homoclinic orbits.
This Research was supported by the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant no. 13A110015).
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