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

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, $$\begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-Δ\right)^s u-λu &= α|u|^{p-2}u + β|u|^{2^*-2}u \quad\mbox{in}\quad Ω,\\ u&=0\quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{cases} \end{align*}$$ where $(-\Delta {)}^s$ is the fractional Laplace operator, $s\in (0,1)$, with $N\ge 3s$, $2<p<2^{\ast }$, $\beta >0,\lambda ,\alpha \in \mathbb{R}$ and establish the existence of nontrivial solutions and sign-changing solutions for the problem $(\mathcal{P})$.