Homoclinic orbits of sub-linear Hamiltonian systems with perturbed terms

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.