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 $$ (-Δ)^{α/ 2} u + \left( V(x) - \fracμ{|x|^α} \right) u = f(x,u)-K(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N \setminus \{0\}, $$ where $V$ is bounded and close-to-periodic potential and $- \fracμ{|x|^α}$ is a Hardy-type potential. We assume that $V$ is positive and $f$ has the subcritical growth but not higher than $|u|^{q-2}u$. 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 $μ\to 0^+$ and $K \to 0$.