Groundstates of the Choquard equations with a sign-changing self-interaction potential

TL;DR

This paper proves the existence of groundstate solutions for a nonlinear Choquard equation with a sign-changing, unbounded self-interaction potential using a relaxed minimization approach.

Contribution

It introduces a method to establish groundstate solutions for Choquard equations with unbounded, sign-changing potentials, covering specific cases like the Newton kernel.

Findings

Existence of nontrivial groundstate solutions under certain conditions.

Convergence of relaxed solutions to the original problem's groundstate.

Applicable to cases with Newton kernels in 1D and 2D.

Abstract

We consider a nonlinear Choquard equation $$ -\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text{in }\mathbb{R}^N, $$ when the self-interaction potential $V$ is unbounded from below. Under some assumptions on $V$ and on $p$, covering $p =2$ and $V$ being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution $u\in H^1 (\mathbb{R}^N)\setminus\{0\}$ by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation.