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 $$(-\Delta {)}^{\alpha /2}u+\left(V(x)-\frac{\mu }{|x{|}^{\alpha }}\right)u=f(x,u)-K(x)|u{|}^{q-2}uon{\mathbb{R}}^N\setminus \{0\},$$ where $V$ is bounded and close-to-periodic potential and $-\frac{\mu }{|x{|}^{\alpha }}$ 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 $\mu$ 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 $\mu$ is negative we show that there is no ground state solutions. We are also interested in an asymptotic behaviour of solutions as $\mu \to 0^{+}$ and $K\to 0$.