Existence and classification of solutions for singular 1-Laplacian equations with gradient terms
Abstract
We study the existence and qualitative properties of non-negative solutions to singular 1-Laplacian problems of the form −Δ1 u = g(u)|D u| + h(u)f & in Ω,newline u=0 & on ∂Ω, where Ω⊂ ℝN is a bounded domain with Lipschitz boundary, 0<f∈ LN,∞(Ω), and g,h are continuous positive functions that may blow up at the origin. We establish the existence of a positive bounded solution even when g is non-integrable at zero and h(0)=+∞, breaking the non-existence phenomenon appearing in the associated p-Laplacian problems. Furthermore, we prove that solutions possess a non-zero boundary trace on a set of positive measure of ∂Ω whenever g(s) grows sufficiently fast near s=0; by contrast, when this growth condition fails, solutions vanishing on the boundary may appear. Finally, when h(0)<+∞, we provide a sharp threshold classification of trivial and non-trivial solutions depending on the size of f.
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BibTeX
@misc{aparicio2026existence,
title = {{Existence and classification of solutions for singular 1-Laplacian equations with gradient terms}},
author = {Antonio J. Martínez Aparicio and Francescantonio Oliva and Francesco Petitta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22027},
eprint = {2610.22027},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22027},
}
RIS
TY - UNPB
TI - Existence and classification of solutions for singular 1-Laplacian equations with gradient terms
AU - Antonio J. Martínez Aparicio
AU - Francescantonio Oliva
AU - Francesco Petitta
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22027
UR - https://arxiv.org/abs/2610.22027
AB - We study the existence and qualitative properties of non-negative solutions to singular 1-Laplacian problems of the form −Δ_1 u = g(u)|D u| + h(u)f & in Ω,newline u=0 & on ∂Ω, where Ω⊂ ℝ^N is a bounded domain with Lipschitz boundary, 0<f∈ L^N,∞(Ω), and g,h are continuous positive functions that may blow up at the origin. We establish the existence of a positive bounded solution even when g is non-integrable at zero and h(0)=+∞, breaking the non-existence phenomenon appearing in the associated p-Laplacian problems. Furthermore, we prove that solutions possess a non-zero boundary trace on a set of positive measure of ∂Ω whenever g(s) grows sufficiently fast near s=0; by contrast, when this growth condition fails, solutions vanishing on the boundary may appear. Finally, when h(0)<+∞, we provide a sharp threshold classification of trivial and non-trivial solutions depending on the size of f.
ER -