Existence and nonexistence of nontrivial solutions for Choquard type equations

Abstract

In this article, we consider the nonlocal problem $$ -\Delta u+u=q(x)\Big(\int_{\mathbb{R}^N}\frac{q(y)|u(y)|^p}{|x-y|^{N-\alpha}}dy \Big)|u|^{p-2}u,\quad x\in \mathbb{R}^N, $$ where $N\geq 3$, $\alpha\in (0,N)$, $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}$ and q(x) is a given potential. Under suitable assumptions on q(x), we prove the existence and nonexistence of nontrivial solutions.

Keywords

• Choquard equation
• nonlocal nonlinearities
• variational methods