Solutions for subquadratic fractional Hamiltonian systems without coercive conditions

In this paper, we are concerned with 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 ) ,where α  ∈ (1 ∕ 2,1), t ∈ R , u ∈ R n , L ∈ C R , Rn 2is a symmetric and positive definite matrix for all t ∈ R , W ∈ C 1R × R n , R , and ∇  W is the gradient of W at u . The novelty of this paper is that, assuming L is bounded in the sense that there are constants 0 <  τ 1  <  τ 2  < + ∞ such that τ 1  |  u  |  2  ≤ ( L ( t ) u , u ) ≤  τ 2  |  u  |  2 for all ( t , u ) ∈ R × R nand 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 [Z. Zhang and R. Yuan, Variational approach to solutions for a class of fractional Hamiltonian systems, Math. Methods Appl. Sci., DOI:10.1002/mma.2941] are generalized and significantly improved. Copyright © 2014 John Wiley & Sons, Ltd.