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

Abstract

We study the existence of solution for the following class of nonlocal problem, $$ -Δu +V(x)u =\Big( I_μ\ast F(x,u)\Big)f(x,u) \quad \mbox{in} \quad \mathbb{R}^2, $$ 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.

References in corpus (1)

• Theory of Bose-Einstein condensation in trapped gases