Concentration phenomena for fractional elliptic equations involving exponential critical growth

TL;DR

This paper studies fractional elliptic equations with exponential critical growth, demonstrating the existence of localized solutions concentrating at potential minima as a small parameter tends to zero.

Contribution

It constructs bound state solutions for a fractional elliptic problem with exponential critical growth, focusing on concentration phenomena at potential minima.

Findings

Existence of localized bound state solutions

Solutions concentrate at potential minima as epsilon approaches zero

Analysis of fractional elliptic equations with exponential growth

Abstract

In this paper, we deal with the following singular perturbed fractional elliptic problem $ \epsilon^{} (-\Delta)^{1/2}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}, $ where $ (-\Delta)^{1/2}u$ is the square root of the Laplacian and $f(s)$ has exponential critical growth. Under suitable conditions on $f(s)$, we construct a localized bound state solution concentrating at an isolated component of the positive local minimum points of the potential of $V$ as $\epsilon$ goes to $0.$