Ground state solutions for a nonlocal equation in ${\mathbb{R}}^2$ involving vanishing potentials and exponential critical growth

Abstract

In this paper, we study the following class of nonlinear equations: $$-\Delta u+V(x)u=\left[|x{|}^{-\mu }\ast (Q(x)F(u))\right]Q(x)f(u),x\in {\mathbb{R}}^2,$$ where $V$ and $Q$ are continuous potentials, which can be unbounded or vanishing at infintiy, $f(s)$ is a continuous function, $F(s)$ is the primitive of $f(s)$, $\ast$ is the convolution operator and $0<\mu <2$. Assuming that the nonlinearity $f(s)$ has exponential critical growth, we establish the existence of ground state solutions by using variational methods. For this, we prove a new version of the Trudinger-Moser inequality for our setting, which was necessary to obtain our main results.

32 pages