A uniqueness result for two-step indefinite p-Laplacian equations
Abstract
Boscaggin, Feltrin and Zanolin conjectured that the Neumann and periodic problems associated with the equation u''+a(t)uγ=0 with a two-step indefinite weight a(t) have at most one positive solution for every γ∈ℝ∖{−1,0,1}. We prove this conjecture and complete the uniqueness picture by also including the logarithmic potential case γ=−1. More generally, for every p>1, we consider the equation (lvert u'rvertp−2u')'+a(t)uγ=0 under Neumann or periodic boundary conditions and establish a complete existence and uniqueness classification for all γ∈ℝ∖{0,p−1}. The necessary mean-value condition γ∫0Ta(t) d t<0 is also sufficient for existence when γ≤−1 or γ>p−1, while an additional sharp condition on the mean of the weight is required in the intermediate range γ∈(−1,p−1)∖{0}. Finally, a numerical example provides evidence of multiple positive solutions for the Neumann problem whose weight has the same single change of sign but a non-constant internal profile, highlighting the role of the two-step assumption.
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BibTeX
@misc{cagnetta2026uniqueness,
title = {{A uniqueness result for two-step indefinite $p$-Laplacian equations}},
author = {Alberto Cagnetta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06068},
eprint = {2610.06068},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06068},
}
RIS
TY - UNPB
TI - A uniqueness result for two-step indefinite p-Laplacian equations
AU - Alberto Cagnetta
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06068
UR - https://arxiv.org/abs/2610.06068
AB - Boscaggin, Feltrin and Zanolin conjectured that the Neumann and periodic problems associated with the equation u''+a(t)u^γ=0 with a two-step indefinite weight a(t) have at most one positive solution for every γ∈ℝ∖{−1,0,1}. We prove this conjecture and complete the uniqueness picture by also including the logarithmic potential case γ=−1. More generally, for every p>1, we consider the equation (lvert u'rvert^p−2u')'+a(t)u^γ=0 under Neumann or periodic boundary conditions and establish a complete existence and uniqueness classification for all γ∈ℝ∖{0,p−1}. The necessary mean-value condition γ∫_0^Ta(t) d t<0 is also sufficient for existence when γ≤−1 or γ>p−1, while an additional sharp condition on the mean of the weight is required in the intermediate range γ∈(−1,p−1)∖{0}. Finally, a numerical example provides evidence of multiple positive solutions for the Neumann problem whose weight has the same single change of sign but a non-constant internal profile, highlighting the role of the two-step assumption.
ER -