Choquard Equations with Mixed Potential

TL;DR

This paper investigates the existence of solutions to a class of nonlinear Choquard equations with mixed potentials, involving convolution-type nonlinearities and variable potential functions, in a multi-dimensional setting.

Contribution

It establishes the existence of nontrivial solutions for Choquard equations with mixed potentials under new assumptions on the potential function.

Findings

Existence of nontrivial solutions proved

Conditions on the potential function are specified

Results extend previous work to mixed potential scenarios

Abstract

In this paper, we study the following class of nonlinear Choquard equation, $$-\Delta u+a(z)u=K(u)f(u)\quad \text{in}\quad \R^N,$$ where $\R^N=\R^L\times\R^M$, $L\geq2$, $K(u)=|.|^{-\gamma}*F(u)$, $\gamma\in(0,N)$, $a$ is a continuous real function and $F$ is the primitive function of $f$. Under some suitable assumptions mixed on the potential $a$. We prove existence of a nontrivial solution for the above equation.