Normalized positive solutions for Schrödinger equations with potentials in unbounded domains

The paper deals with the existence of positive solutions with prescribed L2 norm for the Schrödinger equation (Formula Presented), where Ω = RN or RN \ Ω is a compact set, ρ > 0, V ≽ 0 (also V ≡ 0 is allowed), p ∈ (2, 2 + N4 ). The existence of a positive solution ū is proved when V verifies a suitable decay assumption (Dρ), or if ⃦V ⃦Lq is small, for some q ≽ N2 (q > 1 if N = 2). No smallness assumption on V is required if the decay assumption (Dρ) is fulfilled. There are no assumptions on the size of RN \ Ω. The solution ū is a bound state and no ground state solution exists, up to the autonomous case V ≡ 0 and Ω = RN