Existence and concentration of ground state solutions for a critical   nonlocal Schr\"odinger equation in $\R^2$
  We study the following singularly perturbed nonlocal Schr\"{o}dinger equation $$ -\vr^2\Delta u +V(x)u =\vr^{\mu-2}\Big[\frac{1}{|x|^{\mu}}\ast F(u)\Big]f(u) \quad \mbox{in} \quad \R^2, $$ where $V(x)$ is a continuous real function on $\R^2$, $F(s)$ is the primitive of $f(s)$, $0<\mu<2$ and $\vr$ is a positive parameter. Assuming that the nonlinearity $f(s)$ has critical exponential growth in the sense of Trudinger-Moser, we establish the existence and concentration of solutions by variational methods. 
Claudianor O. Alves, Daniele Cassani, Cristina Tarsi, Minbo Yang
math.AP