Ground states for a fractional scalar field problem with critical growth

TL;DR

This paper proves the existence of a ground state solution for a fractional scalar field equation involving the fractional Laplacian with critical growth in an unbounded domain.

Contribution

It establishes the existence of ground states for a fractional scalar field problem with critical growth, extending previous results to fractional operators.

Findings

Existence of ground state solutions proven.

Applicable to equations with critical growth.

Utilizes variational methods for fractional operators.

Abstract

We prove the existence of a ground state solution for the following fractional scalar field equation $(-\Delta)^{s} u= g(u)$ in $\mathbb{R}^{N}$ where $s\in (0,1), N> 2s$,$ (-\Delta)^{s}$ is the fractional Laplacian, and $g\in C^{1, \beta}(\mathbb{R}, \mathbb{R})$ is an odd function satisfying the critical growth assumption.