Normalized Solutions for Schrödinger Equations with General Nonlinearities on Bounded Domains

We consider the existence of normalized solutions of the following nonlinear Schrödinger equation \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=g(u)\ \text { in }\Omega , & \\ \displaystyle \int _{\Omega }\left| u\right| ^{2}dx=\rho ^{2} \text {, }u\in H_{0}^{1}(\Omega ), & \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+\lambda u=g(u)\text{in}\mathrm{\Omega },\hfill & \\ {\displaystyle {\int }_{\mathrm{\Omega }}{\left|u\right|}^{2}dx={\rho }^2\text{,}u\in H_0^1(\mathrm{\Omega }),}\hfill \end{array}\right)\end{array}$$ where \(\Omega \subset \mathbb {R}^{N}\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary, \(N\ge 3\) $N\ge 3$ , 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 \(\Omega \) $\mathrm{\Omega }$ is a star-shaped domain in \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ . Moreover, we uncover a relation between normalized solutions in \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ and the corresponding normalized solutions on bounded domain \(B_{R}(0)\) $B_R(0)$ by analyzing the behavior of these solutions as \(R\rightarrow \infty \) $R\to \infty$ . Our results are more general and we propose a different variational approach to deal with nonlinear Schrödinger equations on bounded domains with prescribed \(L^{2}\) $L^2$ -norm.