Critical p-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality
Abstract
We investigate positive weak solutions of the critical p-Laplace equation −Δp u = up*−1 h(u), 1 < p < n, where h is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation in a compact family determined by h: Any such solution must coincide with an Aubin--Talenti profile. Moreover, the existence of an Aubin--Talenti profile as a solution implies that h is constant on the entire interval of values attained by that profile. Subsequently, applying this classification result, we establish the following scale-invariant Schoen-type estimate (supBR u)(infB2R u)p−1 ≤ C Rp−n for nonnegative weak solutions defined in B3R. As a direct corollary, we apply the profound result of Ciraolo--Gatti \cite{CG2026} to obtain a fully unrestricted Liouville theorem: Every positive entire solution must coincide with an Aubin--Talenti profile. For the purely critical equation, we also show that the corresponding Liouville classification is equivalent to a Schoen-type Harnack inequality. The arguments in this work give quasilinear versions of the Kelvin transform and the method of moving spheres, which was previously available only in the semilinear setting, and also yield alternative proofs of the classical results. The technique developed here can likely be extended to a wider class of Liouville-type problems for critical equations.
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BibTeX
@misc{zhang2026critical,
title = {{Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality}},
author = {Yi Ru-Ya Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.09113},
eprint = {2608.09113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09113},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical p-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality
AU - Yi Ru-Ya Zhang
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2608.09113
UR - https://arxiv.org/abs/2608.09113
AB - We investigate positive weak solutions of the critical p-Laplace equation −Δ_p u = u^p^*−1 h(u), 1 < p < n, where h is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation in a compact family determined by h: Any such solution must coincide with an Aubin--Talenti profile. Moreover, the existence of an Aubin--Talenti profile as a solution implies that h is constant on the entire interval of values attained by that profile. Subsequently, applying this classification result, we establish the following scale-invariant Schoen-type estimate (sup_B_R u)(inf_B_2R u)^p−1 ≤ C R^p−n for nonnegative weak solutions defined in B_3R. As a direct corollary, we apply the profound result of Ciraolo--Gatti \cite{CG2026} to obtain a fully unrestricted Liouville theorem: Every positive entire solution must coincide with an Aubin--Talenti profile. For the purely critical equation, we also show that the corresponding Liouville classification is equivalent to a Schoen-type Harnack inequality. The arguments in this work give quasilinear versions of the Kelvin transform and the method of moving spheres, which was previously available only in the semilinear setting, and also yield alternative proofs of the classical results. The technique developed here can likely be extended to a wider class of Liouville-type problems for critical equations.
ER -