Homoclinic orbits of sub-linear Hamiltonian systems with perturbed terms

*Correspondence: guanweic@163.com 2School of Mathematical Sciences, University of Jinan, Jinan 250022, Shandong Province, P.R. China Full list of author information is available at the end of the article Abstract By using variational methods, we obtain the existence of homoclinic orbits for perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems.


Introduction and the main result
In this paper, we study the existence of homoclinic orbits for the following second order Hamiltonian systems with perturbed terms: -ü(t) + A(t)u(t)λu(t) = χ(t)∇F u(t) + h(t), t ∈ R, (1.1) where u ∈ R N , A(t) is continuous T-periodic N × N symmetric matrix valued function, λ ∈ R, h ∈ R N , F(u) ∈ C 1 (R N , R) and ∇F(u) denotes its gradient with respect to the u variable.
As usual, we say that u(t) is a homoclinic orbit of (1.1) if u(t) is a solution of (1.1) and u(t) ∈ C 2 (R, R N ) such that u(t) → 0 as |t| → ∞.
, the spectrum of -d 2 dt 2 + A(t). Let (·, ·) be the inner product in R N , and the associated norm is denoted by | · |. Assume that By F(0) = 0 and the differential mean value theorem, we have , u for some s ∈ (0, 1).

Variational frameworks and proof of the main result
Let Ebe a separable closed subspace of a Hilbert space E with inner product ·, · and norm · , and E + = (E -) ⊥ . For some R > 0, set Then M is a submanifold of Ewith boundary ∂M. On E we will also use a topology τ generated by the norm where P ± : E → E ± is the orthogonal projection of E onto E ± and {e k } is a total orthonormal sequence in E -. Obviously, Notations We shall denote by · L q and · q the usual L q (R, R N )-norm and L q (R, R)- and is weakly sequentially continuous if u j u implies (u j ) (u).
Next, we shall use the following generalized saddle point theorem to prove our main result.
Under assumption (L 1 ), B := -d 2 dt 2 + A(t)λ is a selfadjoint operator acting on L 2 := L 2 (R, R N ) with domain D(B) = H 2 (R, R N ) and we have the orthogonal decomposition L 2 = L -⊕ L + , u = u -+ u + such that B is negative (resp., positive) in L -(resp., in L + ). Let E := D(|B| 1/2 ) be equipped, respectively, with the inner product and norm where (·, ·) L 2 denotes the inner product of L 2 (R, R N ). Then we have the decomposition orthogonal with respect to both (·, ·) L 2 and ·, · . By (L 1 ), E = H 1 (R, R N ) with equivalent norms. Then E is a Hilbert space and it is not difficult to show that E ⊂ C 0 (R, R N ), the space of continuous functions u on R such that u(t) → 0 as |t| → ∞ (see, e.g., [18]). Therefore, the corresponding functional of (1.1) can be written as where (u) := R [χ(t)F(u) + h(t)u] dt. By assumptions (L 1 ), (X1) and (F1), it is easy to verify that , ∈ C 1 (E, R) and the derivatives are given by going to a subsequence if necessary. Clearly, (X1) and (F1) imply χ(t)F(u) ≥ 0 for all (t, u) ∈ R × R N , which together with (2.5) and Fatou's lemma implies By (2.6), (2.7), (u j ) ≥ C 0 , the definition of and the weak lower semicontinuity of the norm, we get It implies that (u) ≥ C 0 . Therefore, is τ -upper semicontinuous. Now, we prove is weakly sequentially continuous on E. By (2.5) and the definition of , we have It follows from F ∈ C 1 , ϕ ∈ C ∞ 0 (R, R N ) and u j → u in L 2 loc (R, R N ) (by (2.5)) that i.e., is weakly sequentially continuous on E. The proof is finished. Proof Obviously, if χ(t) ≡ 0 (t ∈ R), then assumption (L 1 ) implies that (1.1) becomes to a linear equation and it is easy to see that it has a solution. Therefore, we may assume that χ ∞ = 0. By the Sobolev inequality, there is a constant C 0 > 0 such that
Consequently, up to a subsequence, we may assume that u j u in E. By (2.10) and the fact that is weakly sequentially continuous (see Lemma 2.2), we have 0 = lim j→∞ u j v = (u)v, ∀v ∈ E.
Therefore, u is a homoclinic orbit of (1.1). The fact h(t) ≡ 0 implies the system (1.1) has no trivial solution, i.e., 0 is not a solution of (1.1), thus u is a nontrivial homoclinic orbit of (1.1). The proof is finished.

Conclusion
We obtain the existence of homoclinic orbits for a class of perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems.