In this paper, we consider the existence of solutions for the following nonlinear Schrödinger equation with $L^{2}$ -norm constraint \( \left \{ \textstyle\begin{array}{l@{\quad }l} (-\Delta )^{s} u=\lambda u+\mu |u|^{q-2} u+ |u|^{p-2} u & \text{ in } \Omega , \\ u=0 & \text{ on } \partial \Omega , \\ \int _{\Omega }u^{2} d x=a^{2}, & \end{array}\displaystyle \right . \) where $s\in (0,1)$ , $\mu ,a>0$ , $N\ge 3$ , $2< q< p<2+\frac{4s}{N}$ , $(-\Delta )^{s}$ is the fractional Laplacian operator, $\Omega \subseteq \mathbb{R}^{N}$ is an exterior domain, that is, $\Omega $ is an unbounded domain in $\mathbb{R}^{N}$ with $\mathbb{R}^{N}\backslash \Omega $ non-empty and bounded and $\lambda \in \mathbb{R}$ is Lagrange multiplier, which appears due to the mass constraint $||u||_{L^{2}(\Omega )}= a$ . In this paper, we use Brouwer degree, barycentric functions and minimax method to prove that for any $a > 0$ , there exists a positive solution $u\in H^{s}_{0} (\Omega )$ for some $\lambda <0$ if $\mathbb{R}^{N}\backslash \Omega $ is contained in a small ball.
