Sharp existence and non-existence for singular $1$-Laplace equations with Hardy potentials
Abstract
In this paper, we investigate the singular Dirichlet problem labelabs1 { rclr −Δ1u &=&λ/|x|Sgn (u)+f/uγ& in Ω, [1ex] u &=&0& on ∂Ω, . where Δ1u=div(Du/|Du|) denotes the $1$-Laplacian operator, the parameters are λ∈ R and γ>0, and f is a non-negative function belonging to the Lorentz space LN,∞(Ω). Our main goal is to establish the existence of non-trivial solutions under the sharp restriction λ<N−1, regardless of the magnitude of the datum f or the value of the singular exponent γ>0. We also show that these solutions are globally bounded and that, on the contrary, no solution exists as soon as λ≥ N−1 and f is positive. These results are achieved by rigorously analyzing the asymptotic behavior, as p→ 1+, of the solutions to the approximating p-Laplace problems { rclc −Δpu&=&λ/|x|p|u|p−2u+f/uγ& in Ω, [1ex] u&=&0& on ∂Ω. . Finally, we provide a family of explicit examples designed to illustrate the sharp optimality of our main assumptions.
Related
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BibTeX
@misc{ortiz2026sharp,
title = {{Sharp existence and non-existence for singular $1$-Laplace equations with Hardy potentials}},
author = {Juan Carlos Chata Ortiz and Francesco Petitta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28298},
eprint = {2609.28298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28298},
}
RIS
TY - UNPB
TI - Sharp existence and non-existence for singular $1$-Laplace equations with Hardy potentials
AU - Juan Carlos Chata Ortiz
AU - Francesco Petitta
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28298
UR - https://arxiv.org/abs/2609.28298
AB - In this paper, we investigate the singular Dirichlet problem labelabs1 { rclr −Δ_1u &=&λ/|x|Sgn (u)+f/u^γ& in Ω, [1ex] u &=&0& on ∂Ω, . where Δ_1u=div(Du/|Du|) denotes the $1$-Laplacian operator, the parameters are λ∈ R and γ>0, and f is a non-negative function belonging to the Lorentz space L^N,∞(Ω). Our main goal is to establish the existence of non-trivial solutions under the sharp restriction λ<N−1, regardless of the magnitude of the datum f or the value of the singular exponent γ>0. We also show that these solutions are globally bounded and that, on the contrary, no solution exists as soon as λ≥ N−1 and f is positive. These results are achieved by rigorously analyzing the asymptotic behavior, as p→ 1^+, of the solutions to the approximating p-Laplace problems { rclc −Δ_pu&=&λ/|x|^p|u|^p−2u+f/u^γ& in Ω, [1ex] u&=&0& on ∂Ω. . Finally, we provide a family of explicit examples designed to illustrate the sharp optimality of our main assumptions.
ER -