Normalized Solutions for Schrödinger Equations with General Nonlinearities on Bounded Domains: Normalized Solutions for Schrödinger Equations: Y. Liu, L. Zhao.

We consider the existence of normalized solutions of the following nonlinear Schrödinger equation - Δ u + λ u = g (u) in Ω , ∫ Ω u 2 d x = ρ 2 , u ∈ H 0 1 (Ω) , <graphic mime-subtype="GIF" href="12220_2024_1890_Article_Equ75.gif"></graphic> where Ω ⊂ R N is a bounded domain with smooth boundary, N ≥ 3 , the nonlinearity g is Sobolev subcritical near infinity and at least mass critical growth near zero. We prove the existence of a solution, which is a local minimizer and obtain a second one of mountain pass type if Ω is a star-shaped domain in R N . Moreover, we uncover a relation between normalized solutions in R N and the corresponding normalized solutions on bounded domain B R (0) by analyzing the behavior of these solutions as R → ∞ . Our results are more general and we propose a different variational approach to deal with nonlinear Schrödinger equations on bounded domains with prescribed L 2 -norm. [ABSTRACT FROM AUTHOR]