Year: 2026
Type: preprint
Abstract: We study normalized solutions to the 1-Laplace equation \[ -\mbox{div}\left(\frac{Du}{|Du|}\right)+λ\frac{u}{|u|}=f(u)+|u|^{1^*-2}u,\quad\mbox{in}~ \mathbb{R}^N \] subject to the constraint $\int_{\mathbb{R}^N}|u| \, \mbox{d}x=m$, where $N\geq2$, $1^*=\frac{N}{N-1}$ is the critical Sobolev exponent for the embedding from BV into $L^p$, $λ, m\in\mathbb{R}^+$ and $f:\mathbb{R}\to\mathbb{R}$ is a continuous nonlinear perturbation. We first establish a Pohozaev-type identity for the equation and use it to study the associated critical energy, and then prove the existence of normalized solutions via nonsmooth critical point theory. A key feature of our approach is the proof of the existence of a Palais-Smale sequence for a Lipschitz continuous functional under a Lipschitz continuous constraint. This is significantly different in methodology from previous problems with $C^2$ constraint.
Source: arXiv (Cornell University)
Authors: Li, Zhouxin, Liu, Ruishu
Institutions: Central South University, Shenzhen University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221183061