Existence of sign-changing solution for a problem involving the fractional Laplacian with critical growth nonlinearities

TL;DR

This paper proves the existence of least energy sign-changing solutions for a fractional Laplacian problem with critical growth nonlinearities using constrained minimization on a Nehari manifold.

Contribution

It introduces a new approach to find sign-changing solutions for fractional problems with critical nonlinearities via constrained minimization.

Findings

Established existence of sign-changing solutions.

Applied Nehari manifold constrained minimization.

Extended results to fractional Laplacian with critical growth.

Abstract

We study the existence of least energy sign-changing solution for the fractional equation $(-\Delta)^{s} u=|u|^{2_{s}^{*}-2}u+\lambda f(x,u)$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^{N},$ $u=0$ in $\mathbb{R}^{N}\setminus \Omega$, where $s\in (0,1)$ and $2_{s}^{*}$ is the fractional critical Sobolev exponent. The proof is based on constrained minimization in a subset of Nehari manifold, containing all the possible sign-changing solutions of the equation.