New existence and multiplicity of homoclinic solutions for second order non-autonomous systems

In this paper, we study the second order non-autonomous system ü(t) + Au̇(t)− L(t)u(t) +∇W(t, u(t)) = 0, ∀t ∈ R, where A is an antisymmetric N × N constant matrix, L ∈ C(R, RN×N) may not be uniformly positive definite for all t ∈ R, and W(t, u) is allowed to be sign-changing and local superquadratic. Under some simple assumptions on A, L and W, we establish some existence criteria to guarantee that the above system has at least one homoclinic solution or infinitely many homoclinic solutions by using the mountain pass theorem or the fountain theorem, respectively. Recent results in the literature are generalized and significantly improved.


Introduction
Consider the following second order non-autonomous system: where u ∈ R N , A is an antisymmetric N × N constant matrix, W ∈ C 1 (R × R N , R), and L ∈ C(R, R N×N ) is a symmetric matrix valued function.As usual, we say that a solution u of system (1.1) is homoclinic to zero if u ∈ C 2 (R, R N ), u = 0, u(t) → 0 and u(t) → 0 as |t| → ∞.
The motivation of our work stems from both theoretical and practical aspects.The importance of homoclinic orbits for dynamical systems has been recognized by Poincaré [14].Thus, the existence and multiplicity of homoclinic solutions has become one of most important problems in the research of dynamical systems.
(1.2) 2 H. W. Chen and Z. M.He The existence and multiplicity of homoclinic solutions for system (1.2) has been intensively studied in many recent papers via variational methods under various hypotheses on L and W; see [1, 4-13, 15, 17-23, 26, 28, 31] and references therein.Most of them treated the case where L(t) and W(t, u) are either independent of t or T-periodic in t; see [4,6,9,11,13,15] and the references therein.In this case, the existence of homoclinic solutions can be obtained by going to the limit of 2kT-periodic solutions of approximating problems.If L(t) and W(t, u) are neither autonomous nor periodic in t, the problem of existence of homoclinic solutions for system (1.2) is quite different from the one just described, because of the lack of compactness of the Sobolev embedding; see for instance [1, 5, 7, 10, 12, 17, 19-23, 26, 28, 31] and the references therein.In [17], Rabinowitz and Tanaka studied system (1.2) without a periodicity assumption for both L and W and obtained the existence of homoclinic solutions for system (1.2) under the Ambrosetti-Rabinowitz growth condition where µ 0 > 2.
Compared with the case where A = 0, the case where A = 0 is more complex.To the best of our knowledge, there are only a few papers that have studied this case; see [25,27,29,30].More precisely, in [27], Yuan and Zhang studied system (1.1) without a periodicity assumption, both for L and W. In detail, they obtained the following results.Theorem 1.1 ([27]).Assume that A, L and W satisfy the following conditions: (A 1 ) L(t) is positive definite symmetric matrix for all t ∈ R and there exist a function l ∈ (A 6 ) W is even in u.
Then system (1.1) has infinitely many homoclinic solutions.
In the present paper, motivated by the above papers, we will study the existence and multiplicity of homoclinic solutions for system (1.1) under more relaxed assumptions on A, L and W.
We will use the following conditions: (H 3 ) W(t, 0) = 0 and there exist c > 0, ν > 2 such that where h 2 , h 3 : R → R + are positive continuous functions such that where for all (t, u) ∈ R × R N , where l 4 , l 5 , l 6 , l 7 : R → R + are positive continuous functions such that where m = min{l(t) : t ∈ R}.
(H 11 ) There exist D > 0 and γ 0 ≥ 2 such that where I N is the unit matrix of order N and f is a continuous bounded function with positive lower bound, and A is an arbitrary antisymmetric N × N constant matrix.It is easy to check that A, L and W satisfying our Theorem 1.3 but not satisfying Theorem 1.1.
Remark 1.7.It is clear that Theorem 1.6 generalizes Theorem 1.1 by relaxing conditions (A 1 )-(A 3 ) and (A 5 ) and removing condition (A 4 ).Let where I N is the unit matrix of order N and f 1 is a continuous bounded function with positive lower bound, and A is an arbitrary antisymmetric N × N constant matrix.It is easy to check that A, L and W satisfying our Theorem 1.6 but not satisfying Theorem 1.1.
Remark 1.13.Obviously, Theorem 1.12 treats the local superquadratic case and Theorem 1.1 just treats the global superquadratic case.Hence, Theorem 1.12 generalizes Theorem 1.1 by relaxing conditions (A 1 )-(A 3 ) and (A 5 ) and removing condition (A 4 ).Furthermore, there are many functions W satisfying our Theorem 1.12 and not satisfying Theorem 1.1.For example, let W(t, u) = f 2 (t) |u| 6 + 6|u| 2 ln(4+|u|) , where The remainder of this paper is organized as follows.In Section 2, some preliminary results are presented.In Section 3, we give the proof of Theorems 1.3, 1.6, 1.9 and 1.12.

Preliminaries
In this section, the following theorems will be needed in our argument.Assume that E is a Banach space with the norm • and E = j∈N X j , where X j are finite dimensional subspace of E.
Then ϕ possesses an unbounded sequence of critical values.
We will get a critical point of ϕ by using a standard version of the mountain pass theorem.Now we state this theorem precisely.Theorem 2.2 ([2, 16]).Let E be a real Banach space and ϕ ∈ C 1 (E, R) satisfy the Palais-Smale condition.If ϕ satisfies the following conditions: (ii) there exist constants ρ, α > 0 such that ϕ ∂B ρ (0) ≥ α; (iii) there exists e ∈ E \ Bρ (0) such that ϕ(e) ≤ 0; then ϕ possesses a critical value d ≥ α given by where B ρ (0) is an open ball in E of radius ρ around 0, and Before establishing the variational setting for system (1.1), we have the following.
and consider the following new second order non-autonomous system: (2.1) Then system (2.1) is equivalent to system (1.1).It is easy to see that all conditions of Theorem 1.3 (or Theorem 1.6) still hold for A, L and Ŵ provided that those hold for A, L and W. Hence we can assume without loss of generality that (L(t)u, u) ≥ α 1 |u| 2 in (H 1 ).
H. W. Chen and Z. M.He We will present some definitions and lemmas that will be used in the proof of our results.In view of Remark 2.3 (or (A 1 )), we consider the function space and the norm , ∀u ∈ X. (2.3) Then E is a Hilbert space with this inner product, and it is easy to verify that E is contin- for all p ∈ [2, +∞].
Define the functional ϕ on E by (2.5) From the assumptions it follows that ϕ is defined on E and belongs to C 1 (E, R), and one can easily check that for any u, v ∈ E. Furthermore, it is routine to verify that any critical point of ϕ in E is a classical solution of system (1.1) with u(±∞) = 0 = u(±∞) (see [27]).

Lemma 2.4. Assume that L satisfies
Proof.The proof is similar to the proof of [9, Lemma 2.1], and we omit it here.
3 Proof of Theorems 1.3, 1.6, 1.9 and 1.12 Now we give the proof of Theorem 1.3.
Proof of Theorem 1.3.We choose a completely orthonormal basis {e j } of E and define E j := Re j , then Z k and Y k can be defined as that in Section 2. By (A 6 ) and (2.6), we obtain that ϕ ∈ C 1 (E, R) is even.Next we will check that all conditions in Theorem 2.1 are satisfied.
Step 1.We verify condition (G 2 ) in Theorem 2.1.Set [24]).By (2.5), (H 1 )-(H 3 ) and Remark 2.3, we have α 1 , it follows from (H 2 ) and Remark 2.3 that ζ > 0. Since β k → 0 as k → ∞, there exists a positive constant N 0 such that By (3.1) and (3.2), we get Step 2. We verify condition (G 1 ) in Theorem 2.1.We follow the idea of the proof of Theorem 1.1 in [26].Firstly, we claim that there exists σ > 0 such that Since dim Y k < ∞, it follows from the compactness of the unit sphere of Y k that there exists a subsequence, say {v n }, such that v n converges to some v 0 in Y k .Hence, we have v 0 = 1.Since all norms are equivalent in the finite-dimensional space, we have Thus there exist σ 1 , In fact, if not, we have for all positive integers n, which implies that and ) and (3.8), we have for all positive integers n.Let n be large enough such that > 0 for all large n, which is a contradiction to (3.7).Therefore, (3.5) holds.For the σ given in (3.5), let New existence and multiplicity of homoclinic solutions 9 By (3.5), we obtain meas It follows from (H 5 ) that for any M 1 > 0 there exists = (M 1 ) > 0 such that Hence we have for all u ∈ Y k with u ≥ σ .It follows from (H 2 ), (H 6 ), (2.4), (3.11), (3.13) and Remark 2.3 that Thus, we can choose u = ρ k large enough (ρ k > r k ) such that Step 3. We prove that ϕ satisfies the Palais-Smale condition.Let {u n } be a Palais-Smale sequence, that is, {ϕ(u n )} is bounded, and ϕ (u n ) → 0 as n → ∞.We now prove that {u n } is bounded in E. In fact, if not, we may assume by contradiction that u n → ∞ as n → ∞.Let w n := u n u n .Clearly, w n = 1 and there is w 0 ∈ E such that, up to a subsequence, Case 1. w 0 = 0.In view of (2.4), (H 2 ), (H 4 ), Remark 2.3 and the Hölder's inequality, one has (3.16) for some M 5 > 0, where Divided by u n 2 on both sides of (3.25), noting that (3.24) and θ ≥ ν − 1, we have It follows from (3.15) and (3.26) that w 0 = 0.That is a contradiction.Case 2. w 0 = 0.The proof is the same as that in Theorem 1.3, and we omit it here.Therefore, {u n } is bounded in E. Similar to the proof of Theorem 1.3, we can prove that {u n } has a convergent subsequence in E. Hence, ϕ satisfies the Palais-Smale condition.The proof is completed.Now we give the proof of Theorem 1.9.
Firstly, we prove that ϕ satisfies the Palais-Smale condition.Suppose that {u n } ⊂ E such that {ϕ(u n )} be a bounded sequence and ϕ (u n ) → 0 as n → ∞.By (2.4), (A 2 ), (H 9 ) and the Hölder's inequality, we have , we get that {u n } is bounded in E. Similar to the proof of Theorem 1.3, we can prove that {u n } has a convergent subsequence in E. Hence, ϕ satisfies the Palais-Smale condition.ϕ(g(s)), where Hence, there exists u * ∈ E such that ϕ(u * ) = d1 and ϕ (u * ) = 0.
Then u * is a desired classical solution of system (1.1).Since d1 > 0, u * is a nontrivial homoclinic solution.
Now we give the proof of Theorem 1.12.
Proof of Theorem 1.12.By (A 6 ) and (2.6), we obtain that ϕ ∈ C 1 (E, R) is even.Next we will check that all conditions in Theorem 2.1 are satisfied.For any k ∈ N, we can choose k + 1 disjoint open sets {Υ i |i = 0, 1, . . ., k} such that . ., v k can extended to be an orthonormal basis {v n } of E. Define X j := Rv j , then Z k and Y k can be defined as that in Section 2.
Step 1.We verify condition (G 2 ) in Theorem 2.1.The proof is similar to the proof of Step 1 in Theorem 1.3.
Step 2. We prove that ϕ satisfies the Palais-Smale condition.The proof is the same as that the proof of Theorem 1.9.