FRACTIONAL HAMILTONIAN SYSTEMS WITH POSITIVE SEMI-DEFINITE MATRIX

We study the existence of solutions for the following fractional Hamiltonian systems {-D-t(infinity)alpha ((-infinity)D(t)(alpha)u(t))( )- lambda L(t)u(t) + del W (t), u(t) = 0, (FHS)(lambda)( ) u is an element of H-alpha (R, R-n), where alpha is an element of (1/2,1), t is an element of R, is an element of R-n, lambda > 0 is a parameter, L is an element of C(R, R-n2) is a symmetric matrix, W is an element of C-1 (R x R-n, R). Assuming that L(t) is a positive semi-definite symmetric matrix, that is, L(t) 0 is allowed to occur in some finite interval T of R, W(t,u) satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)(lambda) has a solution which vanishes on R \ T as lambda -> infinity, and converges to some (u) over tilde is an element of H-alpha (R, R-n). Here, (u) over tilde is an element of E-0(alpha) is a solution of the Dirichlet BVP for fractional systems on the finite interval T. Our results are new and improve recent results in the literature even in the case alpha = 1.