Normalized solutions of mass subcritical Schrödinger equations in exterior domains

In this paper, we study the nonlinear Schrödinger equation with L 2 -norm constraint: - Δ u = λ u + | u | p - 2 u in Ω , u = 0 on ∂ Ω , ∫ Ω | u | 2 d x = a 2 , where N ≥ 3 , Ω ⊆ R N is an exterior domain, i.e., Ω is an unbounded domain with R N \ Ω ¯ non-empty and bounded, a > 0 , 2 < p < 2 + 4 N , and λ ∈ R is Lagrange multiplier, which appears due to the mass constraint ‖ u ‖ L 2 ( Ω ) = a . We use Brouwer degree, barycentric functions and minimax method to prove that for any a > 0 , there is a positive solution u ∈ H 0 1 ( Ω ) for some λ < 0 if R N \ Ω is contained in a small ball. In addition, if we remove the restriction on Ω but impose that a > 0 is small, then we also obtain a positive solution u ∈ H 0 1 ( Ω ) for some λ < 0 . If Ω is the complement of unit ball in R N , then for any a > 0 , we get a positive radial solution u ∈ H 0 1 ( Ω ) for some λ < 0 by Ekeland variational principle. Moreover, we use genus theory to obtain infinitely many radial solutions { ( u n , λ n ) } with λ n < 0 , I p ( u n ) < 0 for n ≥ 1 and I p ( u n ) → 0 - as n → ∞ , where I p is the corresponding energy functional.