We study the existence of two positive normalized solutions for the following fractional critical Schrödinger equation: \(\begin{aligned} \left\{ \begin{aligned}&(-\Delta )^{s} u = \lambda u + |u|^{2_{s}^{*}-2}u + a |u|^{p-2} u, {\text { in }} \Omega ,\\&u > 0 \ {\text { in }} \Omega , \ \ u = 0 \ {\text { on }} \mathbb {R}^N {\setminus } \Omega , \\&\int _{\Omega } {|u|^2}\,\textrm{d} x = c, \end{aligned}\right. \end{aligned}\) where \(\Omega \) is a bounded, smooth and star-shaped domain in \(\mathbb {R}^N \) , \(N > 2s\) , \(s \in \left( 0, 1 \right) \) , \(c > 0 \) and \(2_{ s}^{*}:= \frac{2N}{N-2s}\) . \(\lambda \in \mathbb {R}\) appears as an unknown Lagrange multiplier, and a, p are in one of the following three cases: \(\begin{aligned} (1) a=0, \ (2) a > 0 , 2 + \frac{4\,s}{N}< p< 2_s^*, \ (3) a< 0 , 2< p < 2_s^*. \end{aligned}\) We firstly establish the existence of a normalized ground state solution. Then, using some novel ideas, we obtain the second positive normalized solution, which is of mountain pass type.
