On nonlinear fractional Schrödinger equations with indefinite and Hardy potentials

Abstract

This paper is concerned with a class of fractional Schrödinger equation with Hardy potential ( − Δ ) s u + V ( x ) u − κ | x | 2 s u = f ( x , u ) , x ∈ R N , where s ∈ ( 0 , 1 ) and κ ⩾ 0 is a parameter. Under some suitable conditions on the potential V and the nonlinearity f, we prove the existence of ground state solutions when the parameter κ lies in a given range by using the non-Nehari manifold method. Moreover, we investigate the continuous dependence of ground state energy about κ. Finally, we are able to explore the asymptotic behavior of ground state solutions when κ tends to 0.