Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

Original · EN

This article study the fractional Hamiltonian systems eqnarray00 ₜD∞α(₋∞Dₜαu) + λL(t)u = ∇ W(t, u), t∈ R, eqnarray where α∈ (1/2, 1), λ>0 is a parameter, L∈ C(R, Rⁿ× ⁿ) and W ∈ C¹(R × Rⁿ, R). Unlike most other papers on this problem, we require that L(t) is a positive semi-definite symmetric matrix for all t∈ R, that is, L(t) ≡ 0 is allowed to occur in some finite interval I of R. Under some mild assumptions on W, we establish the existence of nontrivial weak solution, which vanish on R I as λ→ ∞, and converge to u in Hα(R); here u ∈ E₀α is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval I.