Choquard equations under confining external potentials

TL;DR

This paper proves the existence of groundstate, infinitely many solutions, and least energy nodal solutions for a nonlinear Choquard equation with confining potentials, using weighted compact embeddings and a Pohozaev identity.

Contribution

It establishes sharp conditions for solution existence of the Choquard equation under confining potentials, including new existence results and a Pohozaev identity.

Findings

Existence of a groundstate solution.

Existence of infinitely many solutions with unbounded energy.

Existence of least energy nodal solutions for p ≥ 2.

Abstract

We consider the nonlinear Choquard equation $$ -\Delta u+V u=(I_\alpha \ast \vert u\vert ^p)\vert u\vert ^{p-2}u \qquad \text{ in } \mathbb{R}^N $$ where $N\geq 1$, $I_\alpha$ is the Riesz potential integral operator of order $\alpha \in (0, N)$ and $p > 1$. If the potential $ V \in C (\mathbb{R}^N; [0,+\infty)) $ satisfies the confining condition $$ \liminf\limits_{\vert x\vert \to +\infty}\frac{V(x)}{1+\vert x\vert ^{\frac{N+\alpha}{p}-N}}=+\infty, $$ and $\frac{1}{p} > \frac{N - 2}{N + \alpha}$, we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and, when $p \ge 2$ the existence of least energy nodal solution. The constructions are based on suitable weighted compact embedding theorems. The growth assumption is sharp in view of a Poho\v{z}aev identity that we establish.