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

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, $I_μ=\frac{1}{|x|^μ}$, $0<μ<2$ 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.

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