Existence and concentration of ground state solutions for a critical nonlocal Schr\"odinger equation in $\R^2$

TL;DR

This paper proves the existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in two-dimensional space, using variational methods under exponential growth conditions.

Contribution

It establishes the existence and concentration of solutions for a critical nonlocal Schrödinger equation with exponential nonlinearity in 2, a novel result in this context.

Findings

Existence of ground state solutions under critical exponential growth.

Solutions concentrate as the perturbation parameter tends to zero.

Application of variational methods to a nonlocal PDE with critical nonlinearity.

Abstract

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.