The fractional Schrödinger equation with Hardy-type potentials and sign-changing nonlinearities

Abstract

We look for solutions to a fractional Schrödinger equation of the following form (−Δ)α∕2u+V(x)−μ|x|αu=f(x,u)−K(x)|u|q−2uonRN∖{0},where V is bounded and close-to-periodic potential and −μ|x|α is a Hardy-type potential. We assume that V is positive and f has the subcritical growth but not higher than |u|q−2u. If μ is positive and small enough we find a ground state solution, i.e. a critical point of the energy being minimizer on the Nehari manifold. If μ is negative we show that there is no ground state solutions. We are also interested in an asymptotic behaviour of solutions as μ→0+ andK→0.