Positive and sign changing solutions to a nonlinear Choquard equation

We consider the problem $-Δu + W(x)u = ((1/{|x|^α} * |u|^{p}) |u|^{p-2}u, u \in H_{0}^{1}(Ω)$, where $Ω$ is an exterior domain in $\mathbb{R}^{N}$, $N\geq3,$ $α\in(0,N)$, $p\in[2,(2N-α)/(N-2)$, $W$ 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 $Ω$ 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.

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