Normalized Solutions of Fractional Schrödinger Equations with Combined Nonlinearities in Exterior Domains

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.