Positive and sign changing solutions to a nonlinear Choquard equation

Abstract

We consider the problem \[-Δu + W(x)u = ((1/{|x|^α} * |u|^{p}) |u|^{p-2}u, u \in H_{0}^{1}(Ω)\], where $\Omega$ is an exterior domain in ${\mathbb{R}}^N$, $N\ge 3,$ $\alpha \in (0,N)$, $p\in [2,(2N-\alpha )/(N-2)$, $W$ is continuous, ${\inf }_{{\mathbb{R}}^N}W>0,$ and $W(x)$ tends to a positive constant as $|x|$ tends to infinity. Under symmetry assumptions on $\Omega$ and $W$, which allow finite symmetries, and some assumptions on the decay of $W$ at infinity, we establish the existence of a positive solution and multiple sign changing solutions to this problem, having small energy.