Normalized Solutions on large smooth domains to the Schrödinger equation with potential and general nonlinearity: Mass super-critical case
Xiaolu Lin, Zongyan Lv
TL;DR
The paper addresses the existence and multiplicity of normalized solutions to the mass-constrained NLS with a potential and mass-supercritical nonlinearity on large bounded domains and in $\mathbb{R}^N$. It overcomes the failure of Pohozaev-based normalization due to the potential by employing a monotonicity-trick–based variational framework and a mountain-pass geometry on mass-constrained manifolds $S_{r,c}$, obtaining solutions on large domains and passing to the whole space as the domain expands. A priori bounds, positivity of the Lagrange multiplier for small mass, and a concentration-compactness analysis yield normalized solutions in $\mathbb{R}^N$ for small $c$ with a clear decomposition into a core limit plus possible bubbles. The results extend the understanding of normalized solutions with nonconstant potentials in the mass-supercritical regime and provide asymptotic and bifurcation insights that complement recent work on related subcritical/supercritical mixtures.
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 $Ω\subset\mathbb{R}^N$ 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).