Normalized Solutions of Nonhomogeneous Mass Supercritical Schrödinger Equations in Bounded Domains

Abstract

This paper aims to consider normalized solutions of Schrödinger equations in bounded domains, that is, find $(\lambda ,u)\in \mathbb R\times H_0^1(\Omega )$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+\lambda u= |u|^{p-2}u+|u|^{q-2}u\quad &{}\text {in}\ \Omega ,\\ \int _{\Omega }u^2dx=\alpha , \end{array}\right. } \end{aligned}$$

where $\Omega \subset \mathbb R^N$ is a bounded smooth domain, V(x) is the potential, $\alpha >0$ is prescribed, $2<p<2+\frac{4}{N}<q<2^*$. In view of the inherent characteristics of the problem with prescribed mass, there are only few approaches and results on the study of normalized solutions in bounded domains, and the multiplicity of normalized solutions to the problem is completely open so far. In the present paper, for any integer $k\in \mathbb N^+$, we construct k normalized solutions by establishing some special links and using the deformation method on the mass constraint manifold. We further consider the asymptotic behavior of these solutions as mass $\alpha$ tending to 0. Our methods can also be applied to settle another open problem, that is, the multiplicity of normalized solutions of Schrödinger equations with combined mass supercritical nonlinearities ($\sum _{i=1}^m|u|^{q_i-2}u$ with $m\in \mathbb N^+$ and $2+\frac{4}{N}<q_i<2^*$ for any $1\le i\le m$) in bounded domains. Our results are new even for the autonomous case $V(x)\equiv 0$.