Normalized Schrödinger equations with mass-supercritical nonlinearity in exterior domains

We consider the problem m −∆u + λu = |u|p−p−2u, where u ∈ H10 (Ω) satisfies |u|2 = m > 0, λ ∈ R and Ω is a smooth exterior domain. We prove the existence of a positive solution with a constrained Morse index less or equal than N + 1 and λ ≥ 0. We treat both the cases m fixed and RN \ Ω small and Ω fixed and m large.