An improved strong comparison principle for singular p-Laplace equations, with applications
Abstract
We consider positive weak solutions of −Δp u=f(u), with f positive and locally Lipschitz continuous. In the singular case 1<p<2, the Harnack-type inequalities, the strong maximum principle for the linearized operator, the description of the critical set and the strong comparison principle established by Damascelli and Sciunzi [Calc.\ Var.\ Partial Differential Equations \textbf{25} (2006), 139--159] have been available, for more than twenty years, only under the restriction (2N+2)/(N+2)<p<2, a range that shrinks to the empty set as N→∞. We remove the dependence on the dimension and prove all of these results for 3/2<p<2. Since the strong comparison principle is used as a black box throughout the qualitative theory of p-Laplace equations, the improvement propagates. We show in particular that the assumption (2N+2)/(N+2)<p<2 may be replaced by 3/2<p<2 in the resolution of Gibbons' conjecture for −Δpu=f(u) by Esposito, Farina, Montoro and Sciunzi [Math.\ Ann.\ \textbf{382} (2022), 943--974] and in their monotonicity theorem in half-spaces for changing-sign nonlinearities [Calc.\ Var.\ Partial Differential Equations \textbf{61} (2022), art.\ 154]; further applications are given in the body of the paper.
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BibTeX
@misc{le2026improved,
title = {{An improved strong comparison principle for singular $p$-Laplace equations, with applications}},
author = {Phuong Le},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07877},
eprint = {2609.07877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07877},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An improved strong comparison principle for singular p-Laplace equations, with applications
AU - Phuong Le
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07877
UR - https://arxiv.org/abs/2609.07877
AB - We consider positive weak solutions of −Δ_p u=f(u), with f positive and locally Lipschitz continuous. In the singular case 1<p<2, the Harnack-type inequalities, the strong maximum principle for the linearized operator, the description of the critical set and the strong comparison principle established by Damascelli and Sciunzi [Calc.\ Var.\ Partial Differential Equations \textbf{25} (2006), 139--159] have been available, for more than twenty years, only under the restriction (2N+2)/(N+2)<p<2, a range that shrinks to the empty set as N→∞. We remove the dependence on the dimension and prove all of these results for 3/2<p<2. Since the strong comparison principle is used as a black box throughout the qualitative theory of p-Laplace equations, the improvement propagates. We show in particular that the assumption (2N+2)/(N+2)<p<2 may be replaced by 3/2<p<2 in the resolution of Gibbons' conjecture for −Δ_pu=f(u) by Esposito, Farina, Montoro and Sciunzi [Math.\ Ann.\ \textbf{382} (2022), 943--974] and in their monotonicity theorem in half-spaces for changing-sign nonlinearities [Calc.\ Var.\ Partial Differential Equations \textbf{61} (2022), art.\ 154]; further applications are given in the body of the paper.
ER -