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We study the existence of solution for the following class of nonlocalproblem, $$ -\Delta u +V(x)u =\Big( I_\mu\ast F(x,u)\Big)f(x,u) \quad \mbox{in}\quad \mathbb{R}^2, $$ 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 existenceof solutions by using variational methods.

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