Existence of solutions for a nonlocal variational problem in $\mathbb{R}^2$ with exponential critical growth

TL;DR

This paper proves the existence of solutions for a nonlocal variational problem in two-dimensional space with exponential critical growth, using variational methods under periodic potential conditions.

Contribution

It establishes the existence of solutions for a nonlocal PDE with exponential critical growth in , a problem not previously addressed with these specific conditions.

Findings

Existence of solutions proven using variational methods.

Handling of exponential critical growth in .

Application to nonlocal problems with periodic potentials.

Abstract

We study the existence of solution for the following class of nonlocal problem, $$ -\Delta u +V(x)u =\Big( I_\mu\ast F(x,u)\Big)f(x,u) \quad \mbox{in} \quad \mathbb{R}^2, $$ where $V$ is a positive periodic potential, $I_\mu=\frac{1}{|x|^\mu}$, $0<\mu<2$ and $F(x,s)$ is the primitive function of $f(x,s)$ in the variable $s$. In this paper, by assuming that the nonlinearity $f(x,s)$ has an exponential critical growth at infinity, we prove the existence of solutions by using variational methods.