Homoclinic orbits for a class of p-Laplacian systems with periodic assumption

In this paper, by using a linking theorem, some new existence criteria of homoclinic orbits are obtained for the p-Laplacian system d(|u̇(t)|p−2u̇(t))/dt + ∇V (t, u(t)) = f(t), where p > 1, V (t, x) = −K(t, x) + W (t, x).

For the p-Laplacian system (1.1) with f (t) ≡ 0 and K(t, x) ≡ 0 (or ) is a positive definite symmetric matrix), recently, under different assumptions, some results on the existence and multiplicity of periodic solutions, subharmonic solutions and homoclinic solutions have been obtained (for example, see [21,22,23,24,25,26]).In [21], the authors considered the existence of subharmonic solutions for system (1.1) with f (t) ≡ 0 and where L ∈ C(R, R N 2 ) is a positive definite symmetric matrix.Under some reasonable assumptions, they obtained that the system has a sequence of distinct periodic solutions with period k j T satisfying k j ∈ N and k j → ∞ as j → ∞.In [22], the authors considered the existence of homoclinic solutions for system (1.1) with f (t) ≡ 0. They assumed that W is asymptotically p-linear at infinity, K satisfies (K1) and W and K are not periodic in t.In [23]- [26], the authors considered the existence and multiplicity of periodic solutions EJQTDE, 2013 No. 67, p. 3 for system (1.1) with f (t) ≡ 0 and K(t, x) ≡ 0. Motivated by [11,14,17,18], in this paper, we consider the existence of homoclinic orbits for system (1.1) and present some new existence criteria.Next, we state our main results.
Remark 1.1.Theorem 1.3 and Theorem 1.4 show that f can be large when r is large, which is different from Theorem A and Theorem B.Moreover, in Theorem 1.1 and Theorem 1.2, if r ∈ (1, +∞), it is also possible that f can be large.

Preliminaries
Similar to [11,14,17,18], we will obtain the homoclinic orbit of system (1.1) as a limit of solutions of a sequence of differential systems: where For p > 1, let L p 2kT (R, R N ) denote the Banach space of 2kT -periodic functions on R with values in R N and the norm defined by On W 1,p 2kT , we define the norm as follows: Then W 1,p 2kT , • E k is a reflexive and uniformly convex Banach space (see [19], Theorem 3.3 and Theorem 3.6).
Lemma 2.1.Let c > 0 and u ∈ W 1,p (R, R N ).Then for every t ∈ R, the following inequalities hold: and Integrating (2.5) on [t − c, t + c] and using the Hölder's inequality, we have The following (2.8) and its proof have been given in [11] (see [11], Lemma 2.2).Here, for readers' convenience, we also present it.In our Lemma 2.2, our main aim is to present the following (2.7) which generalizes Lemma 2.2 in [11] in some sense.
If p = 2 and u ∈ E k , then the following better result holds: It follows from (2.9), (2.10) and Young's inequality that When p = 2, it follows from (2.11) and Young's inequality that If p = 2 and u ∈ E k , then the following better result holds: Remark 2.3.Corollary 2.2 generalizes (3.3) in [11].
It is easy to obtain that ϕ By (H2) or (H2) , for all u ∈ E k , we obtain It is well known that critical points of ϕ correspond to solutions of system (1.1).
3) Γ(1)E is a single point in E and Γ(t)A converges uniformly to Γ(1)E as t → 1 for each bounded set A ⊂ E.
4) For each t 0 ∈ [0, 1) and each bounded set A ⊂ E, Let Φ be the set of all continuous maps Γ as defined above.We use the following theorem to prove our main results.
Lemma 3.3.Under the assumptions of Theorem 1.1, for every k ∈ N, system (2.1) has a nontrivial solution u k in E k .
Proof.We first construct A and B which satisfy assumptions in Theorem 2.1.
(i) when r ∈ (0, 1], by Corollary 2.1, (H1), (H3)(i), Hölder inequality and γ < p, for (H6)(i) implies that there exists α > 0 such that (ii) when r ∈ (1, +∞), by Corollary 2.1, (H1), Hölder's inequality and γ < p, for (H6)(ii) implies that there exists α > 0 such that EJQTDE, 2013 No. 67, p. 14 By Lemma 3.1 and the periodicity of K, there exists a constant B 0 > 0 such that where By (H4), we know that there exist ε 0 > 0 and L > 0 such that By (3.4) and the periodicity of W , there exists a constant B 1 > 0 such that Since K(t, 0) ≡ 0 and W (t, 0) ≡ 0 which is implied by (H5), we have for all ξ ∈ R. Then by (3.5), we have So there exists ξ 0 ∈ R such that ξ 0 w k > r C 0 and ϕ(ξ 0 w k ) < 0.Moreover, it is clear that ϕ k (0) = 0. Let e 1 = ξ 0 w k and and there exists a sequence {u n } ⊂ E k such that Then there exists a constant C 1k > 0 such that It follows from (H5) and the periodicity and continuity of W that So by (3.5), there exists Hence, it follows from (H2), (3.8) and (3.10) that Note that By assumption (V) and (3.16), we have Since f k (t) is bounded, (3.16) also implies that On the other hand, it is easy to derive from (3.16) and the boundedness of {u n } that kT −kT Set Then we have EJQTDE, 2013 No. 67, p. 18 and From (3.22) and (3.23), we obtain On the other hand, it follows from (3.15) that By (3.24), (3.25) and the Hölder's inequality, we get which, together with (3.26) and (3.27) yields u n E k → u k E k (see [10]).By the uniform convexity of E k and (3.15), it follows from the Kadec-Klee property (see [27]) that u n − EJQTDE, 2013 No. 67, p. 19 u k E k → 0.Moreover, by the continuity of ϕ k and ϕ k , we obtain ϕ k (u k ) = 0 and It is clear that u k = 0 and so u k is a desired nontrivial solution of system (2.1).The proof is complete.
Lemma 3.4.Let {u k } k∈N be the solution of system (2.1).Then there exists a subsequence Proof.First, we prove that the sequence {c k } k∈N is bounded and the sequence {u k } k∈N is uniformly bounded.Second, we prove { uk } k∈N is also uniformly bounded.Finally, we prove both {u k } and { uk } are equicontinuous and then by using the Arzelà-Ascoli Theorem, we obtain the conclusion.We only prove the first step.The rest of proof is the same as Lemma 3.2 in [17].For every Then Γ ∈ Φ.Note that set A = {0, e 1 }.So (3.7) implies that where M 0 is independent of k ∈ N.Moreover, ϕ k (u k ) = 0. Then it follows from (H2) and (3.10) that Then Note that γ > ν + 1.So (H6) implies there exists M 1 > 0 (independent of k) such that Thus the proof is complete.
Proof.The proof is the same as Step 1-Step 3 in the proof of Lemma 3.3 in [17].
Proof of Theorem 1.2.The proof is easy to be completed by replacing in the proofs of Lemma 3.3 and Lemma 3.4.
Proofs of Theorem 1.3 and Theorem 1.4.We only note that in the proof of Lemma 3.3, when γ = p, we dot not need r ∈ (0, 1] and it is sufficient that r > 0. The remaining parts of the proofs are the same as the proofs of Theorem 1.1 and Theorem 1.2, respectively. Proof of Theorem 1.5.Note that f ≡ 0. By (H1), (H3) and γ < p, for u ∈ E k with So (H3) implies that there exists α > 0 such that (H5) implies that W (t, 0) ≡ 0 and (H2) implies that (H2).So (3.6) holds with f 1 (t) ≡ 0.
Hence there exists ξ 0 ∈ R such that ξ 0 w k > r C 0 and ϕ(ξ 0 w k ) < 0.Moreover, it is clear that ϕ k (0) = 0. Let Similar to the argument in Lemma 3.3 and Lemma 3.4 with f (t) ≡ 0, noting that it is sufficient ν < γ < p when f ≡ 0, we can obtain that u k is a desired nontrivial solution of system (2.1).By the Step 1-Step 3 in the proof of Lemma 3.3 in [17], we obtain that u 0 (t) → 0 and u0 (t) → 0 as t → ±∞.Next, we prove, when f ≡ 0, u 0 is nontrivial.The proof is the similar to that in [18] and same as step 4 in the proof of Lemma 3.3 in [17] (with γ = p and b = a there).Here, for readers' convenience, we also present it.It is easy to see that the function Y defined in (H7) is continuous, nondecreasing, Y (s) ≥ Y (0) ≥ 0.
By the definition of Y , we have Integrating the above inequality on the interval [−kT, kT ], we obtain that for every k ∈ N, Then The remainder of the proof is the same as in [7,11,17,18].If u k L ∞ 2kT → 0 as k → ∞, we would have Y (0) ≥ min{1, a}, a contradiction to (H7).Thus there is m > 0, which is independent of k, such that Proof of Theorem 1.6.Similar to the argument of Lemma 3.3 and Lemma 3.4, it is easy to obtain that, under the conditions of Theorem 1.6, u k is a desired nontrivial solution of system (2.1).Then by the proof of Theorem 1.5, we know that u 0 is nontrivial.

Definition 2 . 1 .
(see[20], Definition 3.2) We say that A links B[hm] if A and B are subsets of E such that A ∩ B = ∅, and for each Γ ∈ Φ, there is a t ∈ (0, 1] such thatΓ(t )A ∩ B = ∅.Example 1. (see [20], page 21) Let B be an open set in E, and let A consist of two points e 1 , e 2 with e 1 ∈ B and e 2 ∈ B. Then A links ∂B[hm].
.12) EJQTDE, 2013 No. 67, p. 16 [10] implies that u n E k is bounded.Similar to the argument of Lemma 2 in[10], next we prove that in E k , {u n } has a convergent subsequence, still denoted by {u n }, such that u n → u k , as n → ∞.Since W1,p2kT is a reflexive Banach space, then there is a renamed subsequence {u n } such that