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Normalized solutions to 1-Laplace equations with critical growth term
Abstract
We study normalized solutions to the 1-Laplace equation −div(Du/|Du|)+λu/|u|=f(u)+|u|1*−2u, in ℝN subject to the constraint ∫ℝN|u| dx=m, where N≥2, 1*=N/(N−1) is the critical Sobolev exponent for the embedding from BV into Lp, λ, m∈ℝ+ and f:ℝ→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 C2 constraint.
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BibTeX
@misc{li2026normalized,
title = {{Normalized solutions to 1-Laplace equations with critical growth term}},
author = {Zhouxin Li and Ruishu Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12224},
eprint = {2610.12224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12224},
}
RIS
TY - UNPB
TI - Normalized solutions to 1-Laplace equations with critical growth term
AU - Zhouxin Li
AU - Ruishu Liu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12224
UR - https://arxiv.org/abs/2610.12224
AB - We study normalized solutions to the 1-Laplace equation −div(Du/|Du|)+λu/|u|=f(u)+|u|^1^*−2u, in ℝ^N subject to the constraint ∫_ℝ^N|u| dx=m, where N≥2, 1^*=N/(N−1) is the critical Sobolev exponent for the embedding from BV into L^p, λ, m∈ℝ^+ and f:ℝ→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.
ER -