Homoclinic Solutions for Fractional Hamiltonian Systems with Indefinite Conditions

Ziheng Zhang, César E Torres Ledesma, Rong Yuan

In this paper we are concerned with the existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems − t D α ∞ (−∞ D α t u(t)) − L(t)u(t) + ∇W(t, u(t)) = 0, u ∈ H α (R, R n), (FHS) where α ∈ (1/2, 1), t ∈ R, u ∈ R n , L ∈ C(R, R n 2) is a symmetric matrix for all t ∈ R, W ∈ C 1 (R × R n , R) and ∇W(t, u) is the gradient of W(t, u) at u. The novelty of this paper is that, when L(t) is allowed to be indefinite and W(t, u) satisfies some new superquadratic conditions, we show that (FHS) possesses infinitely many homoclinic solutions via a variant fountain theorem. Recent results are generalized and significantly improved.