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This paper studies the nonlocal fractional analog of the famous paper of Brezis and Nirenberg [Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477]. Namely, we focus on the following model: P−Δsu−λu=α|u|p−2u+β|u|2s∗−2uinΩ,u=0inRN∖Ω,$$\begin{align*}\hskip5pc {\left(\mathcal{P}\right)} {\left\{ \def\eqcellsep{&}\begin{array}{l} {\left(-\Delta \right)}^s u-\lambda u = \alpha |u|^{p-2}u + \beta |u|^{2^*_s-2}u \quad \mbox{in}\quad \Omega ,\\ u=0\quad \mbox{in}\quad \mathbb {R}^N\setminus \Omega , \end{array} \right.}\hskip-5pc \end{align*}$$where (−Δ)s$(-\Delta)^s$ is the fractional Laplace operator, s∈(0,1)$s \in (0,1)$, with N>2s$N > 2s$, 20,λ,α∈R$\beta >0,\, \lambda , \alpha \in \mathbb {R}$, and establish the existence of nontrivial solutions and sign‐changing solutions for the problem (P)$(\mathcal{P})$. [ABSTRACT FROM AUTHOR]