Homoclinic Orbit For Second Order Hamiltonian Systems

In this paper, firstly, we consider the following second order Hamiltonian systemswhere L ∈ C¹(R, R^N2) is a symmetric matrix valued function, W ∈ C¹(R×R^N,R) and â–½W(t, x) = ((?)W/(?)x)(t, x). We consider the existence of homoclinic orbits for systems (HS), without any periodic and coercivity assumption, precisely speaking, assuming that L(t) = 0, specially W(t, x) is even on t and satisfies a class of new superquadratic condition which is different from the corresponding conditions in [6], [7], [8], [9], [10], [12], [14], [15], [17], [19], [20], [22], [24], [25] and [26]. It includes the asymptotically quadratic case besides the superquadratic case. When L(t) is uniformly positively definite, we only discuss the superquadratic case under a class of new superquadratic condition which is weaker than Ambrosetti-Rabinowitz condition. We have adopted some ideas from [10] and prove the existence of a homoclinic solution as the limit of the solutions of boundary-value problems, which is done by using the Mountain Pass Theorem to get the solutions of boundary-value problems and approximating estimates to pass to a nontrivial limit.The main results are the following theorems:Theorem 4.1 Assume that W satisfies(W1) W(t,0) ≡ 0, W(-t,x) = W{t,x) for all t∈R and x ∈ R^N;(W2) W_t(t, x) ≤ 0 for all t ≥ 0 and x ∈ R^N;(W3) There exist constants d₁ > 0 and r ≥ 2 such that...