Existence of Solutions for a Nonlocal Variational Problem in R^(2) with Exponential Critical Growth

We study the existence of nontrivial solutions 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 $s$. By assuming that the nonlinearity $f(x,s)$ has an exponential critical growth at infinity, we prove the existence of solutions by variational methods.