Normalized Solutions of Mass Subcritical Fractional Schrödinger Equations in Exterior Domains

In this paper, we look for solutions to the following fractional nonlinear Schrödinger equations with prescribed L^2 -norm constraint: {[ (-Δ )^s u=λ u+|u|^p-2u in Ω,; u=0 on ∂Ω,; ∫ _Ω |u|^2dx=a^2,; ]. where s∈ (0,1) , Ω⊂ℝ^N ( N≥ 3 ) is an exterior domain with smooth boundary ∂Ω∅ such that ℝ^N\Ω is bounded. Combining minimax method, barycentric functions, and Brouwer degree theory, we prove the existence of positive normalized solution for any a>0 , when 20 is small, we can still achieve the same result without any restrictions on Ω . Finally, if Ω is the complement of the unit ball in ℝ^N , with the aid of genus theory, we establish the existence and multiplicity of radial normalized solutions for any a>0 .