Normalized Solutions on large smooth domains to the Schrödinger equation with potential and general nonlinearity: Mass super-critical case

Abstract

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrödinger equation with general nonlinearity: Mass super-critical case: \[\begin{cases} -Δu+V(x)u+λu=g(u),\\ \|u\|_2^2=\int|u|^2\mathrm{d}x=c, \end{cases} \] both on large bounded smooth star-shaped domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ and on ${\mathbb{R}}^N$, where $V(x)$ is the potential and the nonlinearity $g(\cdot )$ considered here are very general and of mass super-critical. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential $V(x)$. In addition, our study can be considered as a complement of Bartsch-Qi-Zou (Math Ann 390, 4813--4859, 2024), which has addressed an open problem raised in Bartsch et al. (Commun Partial Differ Equ 46(9):1729--1756, 2021).

arXiv admin note: substantial text overlap with arXiv:2412.03296; text overlap with arXiv:501.04071,arXiv:2306.07826 by other authors