Variational approach to solutions for a class of fractional Hamiltonian systems

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: FHS{t D ∞ α ( − ∞ D t α u ( t ) )+ L ( t ) u ( t ) = ∇ W ( t , u ( t ) ) ,u ∈ H α ( ℝ , ℝ n ) ,( FHS )where α  ∈ (1 ∕ 2,1), t ∈ R , u ∈ R n , and L ∈ C ( R , R n 2 ) are symmetric and positive definite matrices for all t ∈ R , W ∈ C 1 ( R × R n , R ) , and ∇  W is the gradient of W at u . The novelty of this paper is that, assuming L is coercive at infinity, and W is of subquadratic growth as |  u  | → + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.