Homoclinic orbits for a class of second-order Hamiltonian systems with concave – convex nonlinearities

In this paper, we study the existence of multiple homoclinic solutions for the following second order Hamiltonian systems ü(t)− L(t)u(t) +∇W(t, u(t)) = 0, where L(t) satisfies a boundedness assumption which is different from the coercive condition and W is a combination of subquadratic and superquadratic terms.


Introduction and main results
In this paper, we consider the following second-order Hamiltonian systems ü(t) − L(t)u(t) + ∇W(t, u(t)) = 0, (1.1) where W : R × R N → R is a C 1 -map and L : R → R N 2 is a matrix valued function.We say that a solution u(t) of problem (1.1) is nontrivial homoclinic (to 0) if u ≡ 0, u(t) → 0 as t → ±∞.
The dynamical system is a class of classical mathematical model to describe the evolution of natural status, which have been studied by many mathematicians (see ).It was shown by Poincaré that the homoclinic orbits are very important in study of the behavior of dynamical systems.In the last decades, variational methods and the critical point theorem have been used successfully in studying the existence and multiplicity of homoclinic solutions for differential equations by many mathematicians (see [1, 3-5, 8-17, 19-21, 23, 24, 27-29, 32-41] and the references therein).
In [20], Rabinowitz made use of the periodicity of L(t) and W(t, x) to obtain the existence of nontrivial homoclinic solution for problem (1.1) as the limit of a sequence of periodic solutions.While L(t) and W(t, x) are neither independent of t nor periodic in t, the problem is quite different from the periodic one since the lack of compactness.In order to get the compactness back, Rabinowitz and Tanaka [21] introduced the following coercive condition on L(t).
(L 0 ) L ∈ C(R, R N 2 ) is a symmetric and positively definite matrix for all t ∈ R and there exists a continuous function l : R → R such that l(t) > 0 for all t ∈ R and (L(t)x, x) ≥ l(t)|x| 2 with l(t) → ∞ as |t| → ∞.
With condition (L 0 ), Omana and Willem [16] obtained a new compact embedding theorem and got the existence and multiplicity of homoclinic solutions for problem (1.1).It is obvious that there are many functions which do not satisfy condition (L 0 ).For instance, let L(t) = (4 + arctan t) Id n , where Id n is the n × n identity matrix.
If there is no periodic or coercive assumption, it is difficult to obtain the compactness of the embedding theorem.Therefore, there are only few papers concerning about this kind of situation.In the present paper, we consider the following condition on L(t).
(L) L ∈ C(R, R N 2 ) is a symmetric and positively definite matrix for all t ∈ R and there exist constants 0 < τ 2 < τ 1 such that Condition (L) was introduced by Zhang, Xiang and Yuan in [41].With condition (L), the authors obtained a new compact embedding theorem.In this paper, W is assumed to be of the following form (1. 2) The existence and multiplicity of homoclinic for problem (1.1) with mixed nonlinearities have been considered in some previous works.In 2011, Yang, Chen and Sun [33] showed the existence of infinitely many homoclinic solutions for problem (1.1).In a recent paper [32], Wu, Tang and Wu obtained the existence and nonuniqueness of homoclinic solutions for problem (1.1) with some nonlinear terms which are more general than those in [33].However, condition (L 0 ) is needed in both of above papers.In this paper, we take advantage of condition (L) to study problem (1.1) with concave-convex nonlinearities.Now we state our main results.Theorem 1.1.Suppose that (L), (1.2) and the following conditions hold . Then there exists λ 1 > 0 such that for all λ ∈ (0, λ 1 ), problem (1.1) possesses at least two homoclinic solutions.Remark 1.2.In [9,33,36], the authors also considered the concave-convex nonlinearities, but in [9,33], L(t) was required to satisfy the coercive condition (L 0 ), which is different from condition (L).In [36], only a class of specific nonlinearities was considered and the concave term was assumed to be positive.Theorem 1.3.Suppose that (L), (1.2), (W 1 )-(W 4 ), (W 6 ) and the following condition hold Then problem (1.1) possesses infinitely many homoclinic solutions.
In our proofs, the following critical point theorems are needed.Lemma 1.8 (Lu [13]).Let X be a real reflexive Banach space and Ω ⊂ X a closed bounded convex subset of X. Suppose that ϕ : X → R is a weakly lower semi-continuous (w.l.s.c. for short) functional.If there exists a point x 0 ∈ Ω \ ∂Ω such that Lemma 1.9 (Chang [7]).Suppose that E is a Hilbert space, I ∈ C 1 (E, R) is even with I(0) = 0, and that (Z 1 ) there are constants , α > 0 and a finite dimensional linear subspace X such that I| X ⊥ ∂B ≥ α, where B = {u ∈ E : u ≤ }; We recall that a functional I is said to satisfy the (PS) * condition with respect to

Preliminaries
Furthermore, the corresponding functional of (1.1) is defined by ) be the weighted space of measurable functions u : R → R N under the norm where With condition (L), Lv and Tang obtained the following compact embedding theorem.
Lemma 2.1 (Lv and Tang [14]).Suppose that assumption (L) holds.Then the imbedding of E in The following lemma is a complement to Lemma 2.1 with the case p = 2.
Lemma 2.2 (Yuan and Zhang [37]).Under condition (L), the embedding Then we can prove the following lemma.Lemma 2.3.Suppose that the conditions (W 6 ), (W 8 ), (W 9 ), (W 10 ) hold, then we have ∇W(t, Proof.Assume that u k u in E. By the Banach-Steinhaus theorem and (2.1), there exists We can deduce from (W 10 ) and (2.4) that there exists , Then we can get that ψ ∈ L r i b i (R, R N ), for any i = 1, 2. By (W 6 ) and the definition of η i , we have Furthermore, (W 9 ) and Lemma 2.
Using Lebesgue's dominated convergence theorem, the lemma is proved.
Similar to the proof of Lemma 2.3 in [9], we can see that I ∈ C 1 (E, R) is w.l.s.c. and for any v ∈ E, which implies that Remark 2.5.Similar to Lemma 3.1 in [41], under condition (L), all the critical points of I are homoclinic solutions for problem (1.1).

Proof of Theorem 1.1
The existence of homoclinic solution is obtained by the Mountain Pass Theorem with (C) condition which is stated as follows.
Lemma 3.3.Suppose the conditions of Theorem 1.1 hold, then there exists e 1 ∈ E such that e 1 > 1 and I(e 1 ) ≤ 0, where 1 is defined in Lemma 3.2.
Lemma 3.4.Suppose the conditions of Theorem 1.1 hold, then I satisfies condition (C).
Proof.Assume that {u n } n∈N ⊂ E is a sequence such that {I(u n )} is bounded and which is a contradiction.Hence {u n } is bounded in E. Consequently, there exists a subsequence, still denoted by {u n }, such that u n u in E. Therefore By Remark 2.4, we have It follows from (2.6) that which implies that u n − u → 0 as n → +∞.where Hence, there exists u 0 ∈ E such that Then the function u 0 is a desired homoclinic solution of problem (1.1).Subsequently, we search for the second critical point of I corresponding to negative critical value.
Lemma 3.5.Suppose that the conditions of Theorem 1.1 hold, then there exists a critical point of I corresponding to a negative critical value.
By Lemma 3.2-Lemma 3.5, we can see that I possesses at least two distinct nontrivial critical points.By Remark 2.5, problem (1.1) possesses at least two homoclinic solutions.

Proof of Theorem 1.3
In this section, we will use Lemma 1.9 to prove the existence of infinitely many homoclinic solutions for problem (1.1).Lemma 4.1.Suppose the conditions of Theorem 1.3 hold, then I satisfies (Z 1 ).
Proof.Let {x j } ∞ j=1 be a complete orthonormal basis of E and X k = k j=1 Z j , where Z j = span{x j }.For any q ∈ [2, +∞], we set It is easy to see that h k (q) → 0 as k → ∞ for any q ∈ [2, +∞].
which implies that there exists a k 0 > 0 such that Lemma 4.2.Suppose the conditions of Theorem 1.3 hold, then for any m ∈ N, there exist a linear subspace Xm and r m > 0 such that dim Xm = m and Proof.By (W 2 ), there exist a 0 > 0 and Λ 0 ⊂ Λ such that a 1 (t) > a 0 for all t ∈ Λ 0 with meas(Λ 0 ) > 0. Choose a complete orthonormal basis {e j Since s > 2 > max{r 1 , r 2 }, there exists r m > 0 such that I(u m ) ≤ 0 for all u m ∈ Xm \ B r m , which proves this lemma.Proof.The proof of this lemma is similar to Lemma 3.4.
Proof of Theorem 1.3.By Lemmas 4.1-4.3 and Lemma 1.9, I possesses infinitely many distinct critical points corresponding to positive critical values.The proof of Theorem 1.3 is finished.
Lemma 5.2.Suppose the conditions of Theorem 1.5 hold, then there exists e 2 ∈ E such that e 2 > 3 and I(e 2 ) ≤ 0, where 3 is defined in Lemma 5.1.
Proof.Choose e 3 ∈ C ∞ 0 (−1, 1) such that e 3 = 1.It follows from (W 9 ) that there exists ã > 0 such that a 2 (t) ≥ ã for all t ∈ (−1, 1).We can see that there exist ẽ > 0 and Υ ⊂ (−1, Therefore, there exists η 1 > 0 such that I(η 1 e 3 ) < 0 and η 1 e 3 > 3 .Let e 2 = η 1 e 3 , we can see I(e 2 ) < 0, which proves this lemma.Then there exists a constant M 4 > 0 such that Subsequently, we show that {u n } is bounded in E. Set where ν is defined in (W 12 ).From (W 10 ), we can deduce that G(x) = o(|x| 2 ) as |x| → 0, then there exists for all |x| ≤ ρ 2 .Arguing by contradiction, we assume that u n → +∞ as n → ∞.Set z n = u n u n , then z n = 1, which implies that there exists a subsequence of {z n }, still denoted by {z n }, such that z n z 0 in E and z n → z 0 uniformly on R as n → ∞.The following discussion is divided into two cases.
Lemma 5.4.Suppose that the conditions of Theorem 1.5 hold, then there exists a critical point of I corresponding to negative critical value.
Proof.The proof is similar to Lemma 3.5.
By Lemmas 5.1-5.4,we can deduce that I possesses at least two critical points.Consequently, problem (1.1) possesses at least two homoclinic solutions.
Lemma 6.3.Suppose the conditions of Theorem 1.7 hold, then I satisfies the (PS) * condition.
Proof.The proof is similar to Lemma 5.3.
Proof of Theorem 1.7.By Lemmas 6.1-6.3 and Lemma 1.9, I possesses infinitely many distinct critical points corresponding to positive critical values.The proof of Theorem 1.7 is finished.

(Z 2 )
there is a sequence of linear subspaces Xm , dim Xm = m, and there exists r m > 0 such that I(u) ≤ 0 on Xm \ B r m , m = 1, 2, . . .If, further, I satisfies the (PS) * condition with respect to { Xm | m = 1, 2, . . .}, then I possesses infinitely many distinct critical points corresponding to positive critical values.

Lemma 4 . 3 .
Suppose the conditions of Theorem 1.3 hold, then I satisfies the (PS) * condition.

Lemma 5 . 3 .
Suppose the conditions of Theorem 1.5 hold, then I satisfies the (PS) condition.Proof.Assume that {u n } n∈N ⊂ E is a sequence such that |I(u n )| < ∞ and I (u n ) → 0.