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The present paper studies the non-local fractional analogue of the famouspaper of Brezis and Nirenberg in [4]. Namely, we focus on the following model,$$\begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-\Delta\right)^su-\lambda u &= \alpha |u|^{p-2}u + \beta|u|^{2^*-2}u \quad\mbox{in}\quad\Omega,\\ u&=0\quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{cases}\end{align*}$$ 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 solutionsfor the problem $(\mathcal{P})$.

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