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Counterexamples to Wang's Conjecture
Abstract
In this paper, we give a negative answer to Wang's conjecture. For every n≥ 3, there exist ε>0 and δ∈ (0,1) such that, for all q and λ satisfying n/(n−2)−ε<q≤ n/(n−2), 1/(q−1)· δ<λ≤ 1/(q−1), there exists a bounded Euclidean domain Ω⊂ℝn with principal curvatures bigger than 1 for which the following nonlinear Robin problem admits a nonconstant positive solution: Δu=0, & in Ω, [2mm] (∂ u)/∂ν+λu=uq, & on ∂Ω.
Related
BibTeX
@misc{jiang2026counterexamples,
title = {{Counterexamples to Wang's Conjecture}},
author = {Ruiqi Jiang and Linlin Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13079},
eprint = {2609.13079},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13079},
}
RIS
TY - UNPB
TI - Counterexamples to Wang's Conjecture
AU - Ruiqi Jiang
AU - Linlin Sun
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13079
UR - https://arxiv.org/abs/2609.13079
AB - In this paper, we give a negative answer to Wang's conjecture. For every n≥ 3, there exist ε>0 and δ∈ (0,1) such that, for all q and λ satisfying n/(n−2)−ε<q≤ n/(n−2), 1/(q−1)· δ<λ≤ 1/(q−1), there exists a bounded Euclidean domain Ω⊂ℝ^n with principal curvatures bigger than 1 for which the following nonlinear Robin problem admits a nonconstant positive solution: Δu=0, & in Ω, [2mm] (∂ u)/∂ν+λu=u^q, & on ∂Ω.
ER -