This paper studies the following fractional Schrödinger equation \(\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle (- \Delta )^s u+\lambda u+ |u|^{q-2}u =|u|^{p-2}u \quad \quad \text {in} \ \mathbb { R}^N, \ N\ge 2, \\ \displaystyle \int _{\mathbb {R}^N}|u|^2dx=a>0, \end{array}\right. } \end{aligned}\) where \( s\in (0, 1)\) and \(2< p, q \le 2_s^*:=\frac{2N}{N-2s}\) is the fractional critical Sobolev exponent. By giving different assumptions on \(2<p<q\le 2_s^*\) , we get the existence, non-existence and concentration behavior of solutions for the above problem. It is worth emphasizing that some of the results are new, even for Laplacian operator.
