The fractional Schr\"odinger equation with Hardy-type potentials and sign-changing nonlinearities

Abstract

We look for solutions to a fractional Schr\"odinger equation of the following form $$ (-\Delta)^{\alpha / 2} u + \left( V(x) - \frac{\mu}{|x|^{\alpha}} \right) u = f(x,u)-K(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N \setminus \{0\}, $$ 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 $K \to 0$.