Normalized Solutions on large smooth domains to the Schr\"{o}dinger equation with potential and general nonlinearity: Mass super-critical case

TL;DR

This paper establishes the existence and multiplicity of normalized solutions for a super-critical nonlinear Schrödinger equation with potential on large domains and the whole space, overcoming limitations of traditional methods.

Contribution

It introduces new techniques to find normalized solutions in the super-critical case with potential, extending previous results and addressing open problems.

Findings

Existence of normalized solutions on large domains and ^N.

Multiple solutions under certain conditions.

Overcoming limitations of Pohozaev identity in super-critical case.

Abstract

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schr\"{o}dinger equation with general nonlinearity: Mass super-critical case: \[\begin{cases} -\Delta u+V(x)u+\lambda u=g(u),\\ \|u\|_2^2=\int|u|^2\mathrm{d}x=c, \end{cases} \] both on large bounded smooth star-shaped domain $\Omega\subset\mathbb{R}^N$ 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).