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: $$\begin{aligned} -\Delta u+V(x) u = \left[ |x|^{-\mu }*(Q(x)F(u))\right] Q(x)f(u),\quad x\in \mathbb {R}^2, \end{aligned}$$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), \(*\) is the convolution operation 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 establish a version of the Trudinger–Moser inequality for our setting, which was necessary to obtain our main results.