Positive and sign changing solutions to a nonlinear Choquard equation

TL;DR

This paper proves the existence of positive and multiple sign-changing solutions for a nonlinear Choquard equation in exterior domains, under symmetry and decay conditions on the potential.

Contribution

It introduces new existence results for positive and sign-changing solutions to a nonlinear Choquard equation with symmetry and decay assumptions.

Findings

Existence of positive solutions under symmetry and decay conditions.

Multiple sign-changing solutions with small energy.

Solutions are obtained in exterior domains with specific potential properties.

Abstract

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