Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity

TL;DR

This paper proves the existence of nontrivial solutions for a class of nonlinear Choquard equations with potential functions that vanish at infinity, using a penalization method under certain assumptions.

Contribution

It establishes the existence of solutions for nonlinear Choquard equations with potentials tending to zero at infinity, extending previous results to this case.

Findings

Existence of nontrivial solutions under potential vanishing at infinity

Application of penalization method to nonlinear Choquard equations

Conditions on potential V ensuring solution existence

Abstract

We study the following class of nonlinear Choquard equation, $$ -\Delta u +V(x)u =\Big( \frac{1}{|x|^\mu}\ast F(u)\Big)f(u) \quad \mbox{in} \quad \R^N, $$ where $0<\mu<N$, $N \geq 3$, $V$ is a continuous real function and $F$ is the primitive function of $f$. Under some suitable assumptions on the potential $V$, which include the case $V(\infty)=0$, that is, $V(x)\to 0$ as $|x|\to +\infty$, we prove existence of a nontrivial solution for the above equation by penalization method.