On the existence of multiple solutions for fractional Brezis Nirenberg type equations

TL;DR

This paper investigates the fractional Brezis-Nirenberg problem, establishing conditions for the existence of multiple solutions, including sign-changing ones, for a non-local fractional Laplacian equation with critical and subcritical nonlinearities.

Contribution

It extends classical Brezis-Nirenberg results to the fractional setting, proving the existence of multiple solutions, including sign-changing solutions, for a non-local fractional PDE.

Findings

Existence of nontrivial solutions under certain parameter conditions

Existence of sign-changing solutions

Extension of classical results to fractional Laplacian context

Abstract

The present paper studies the non-local fractional analogue of the famous paper of Brezis and Nirenberg in [4]. Namely, we focus on the following model, $$\begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-\Delta\right)^s u-\lambda u &= \alpha |u|^{p-2}u + \beta|u|^{2^*-2}u \quad\mbox{in}\quad \Omega,\\ u&=0\quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{cases} \end{align*}$$ where $(-\Delta)^s$ is the fractional Laplace operator, $s \in (0,1)$, with $N \geq 3s$, $2<p<2^*$, $\beta>0, \lambda, \alpha \in \mathbb{R}$ and establish the existence of nontrivial solutions and sign-changing solutions for the problem $(\mathcal{P})$.