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

TL;DR

This paper proves the existence of ground state solutions for a class of nonlocal nonlinear equations in two dimensions with potentials that may vanish or be unbounded, using variational methods and a new Trudinger-Moser inequality.

Contribution

It introduces a new version of the Trudinger-Moser inequality tailored for nonlocal equations with exponential critical growth in a72, and establishes ground state solutions under these conditions.

Findings

Existence of ground state solutions for the nonlocal equation.

Development of a new Trudinger-Moser inequality for the setting.

Application of variational methods to nonlinear equations with vanishing potentials.

Abstract

In this paper, we study the following class of nonlinear equations: $$ -\Delta u+V(x) u = \left[|x|^{-\mu}*(Q(x)F(u))\right]Q(x)f(u),\quad x\in\mathbb{R}^2, $$ 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 $f(s)$, $*$ 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.