Extremal parameters, multiplicity and nonexistence for a singular elliptic problem
Abstract
We consider the following class of singular elliptic problems −Δu = −u/(|u|β+1)χ{|u|>0} +λ|u|p−1u, & in Ω, [1mm] u=0, & on ∂Ω, where Ω⊂ℝN is a bounded smooth domain, N≥3, 0<β,p<1, and λ>0. The energy functional is not of class C1 in H01(Ω), and the standard critical point theory does not apply. We prove that a nontrivial nonnegative weak solution exists if and only if λ exceeds a critical value λ̅, which we identify and locate strictly between two Rayleigh-type parameters 0<λ*<λ̅<λ*<+∞. This gives a sharp existence threshold. We also prove that for every λ in a neighborhood on the left of λ*, there exist at least two distinct nontrivial nonnegative weak solutions. The first is a minimizer on the Nehari manifold, with an energy changing sign at λ*. The second has a strictly positive energy. For λ<λ* sufficiently close to λ*, both solutions have positive energy. This contrasts with previous multiplicity results for the same problem, in which the two solutions obtained have energies of opposite signs.
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BibTeX
@misc{albuquerque2026extremal,
title = {{Extremal parameters, multiplicity and nonexistence for a singular elliptic problem}},
author = {J. C. de Albuquerque and M. L. M. Carvalho and Edcarlos D. Silva and M. L. Silva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08389},
eprint = {2610.08389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08389},
}
RIS
TY - UNPB
TI - Extremal parameters, multiplicity and nonexistence for a singular elliptic problem
AU - J. C. de Albuquerque
AU - M. L. M. Carvalho
AU - Edcarlos D. Silva
AU - M. L. Silva
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08389
UR - https://arxiv.org/abs/2610.08389
AB - We consider the following class of singular elliptic problems −Δu = −u/(|u|^β+1)χ_{|u|>0} +λ|u|^p−1u, & in Ω, [1mm] u=0, & on ∂Ω, where Ω⊂ℝ^N is a bounded smooth domain, N≥3, 0<β,p<1, and λ>0. The energy functional is not of class C^1 in H_0^1(Ω), and the standard critical point theory does not apply. We prove that a nontrivial nonnegative weak solution exists if and only if λ exceeds a critical value λ̅, which we identify and locate strictly between two Rayleigh-type parameters 0<λ_*<λ̅<λ^*<+∞. This gives a sharp existence threshold. We also prove that for every λ in a neighborhood on the left of λ^*, there exist at least two distinct nontrivial nonnegative weak solutions. The first is a minimizer on the Nehari manifold, with an energy changing sign at λ^*. The second has a strictly positive energy. For λ<λ^* sufficiently close to λ^*, both solutions have positive energy. This contrasts with previous multiplicity results for the same problem, in which the two solutions obtained have energies of opposite signs.
ER -