Multiplicity of Solutions for Fractional Hamiltonian Systems with Liouville-Weyl Fractional Derivatives

Abstract

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems:

tDα(−∞Dαt u(t)) + L(t)u(t) = ∇W(t, u(t)), (0.1)

u ∈ Hα(ℝ,ℝN),

where α ∈ (1/2, 1), t ∈ ℝ, u ∈ ℝn, L ∈ C(ℝ,ℝn2 ) is a symmetric and positive definite matrix for all t ∈ ℝ, W ∈ C1(ℝ × ℝn,ℝ), and ∇W is the gradient of W at u. The novelty of this paper is that, assuming there exists l ∈ C(ℝ,ℝ) such that (L(t)u, u) ≥ l(t)|u|2 for all t ∈ ℝ, u ∈ ℝn and the following conditions on l: inf t ∈ ℝ l(t) > 0 and there exists r0 > 0 such that, for any M >0

m({t ∈ (y − r0, y + r0)/ l(t) ≤ M}) → 0 as |y| →∞

are satisfied and W is of subquadratic growth as |u| → +∞, we show that (0.1) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in Z. Zhang and R. Yuan [24] are significantly improved.

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Fractional Calculus and Applied Analysis (FCAA) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order.

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