Liouville theorems and new gradient estimates for positive solutions to $\Delta_pu+au^q=0$ on a complete manifold

Youde Wang and Liqin Zhang

Vol. 342 (2026), No. 2, 395–426

DOI: 10.2140/pjm.2026.342.395

Abstract

We use the Saloff-Coste Sobolev inequality and the Nash–Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation ${\mathrm{\Delta }}_{p}u+a{u}^q=0$ defined on a complete Riemannian manifold $(M,g)$ with Ricci lower bound, where $p>1$ is a constant and ${\mathrm{\Delta }}_{p}u=\mathrm{div}<!--FUNCTION\; APPLICATION-->(|\nu {|}^{p-2}\nu )$ is the usual $p$-Laplace operator. Under certain assumptions on $a$, $p$ and $q$, we derive some gradient estimates and Liouville type theorems for positive solutions to the above equation. In particular, under certain assumptions on $a$, $p$ and $q$ we show whether or not the exact Cheng–Yau $\log <!--FUNCTION\; APPLICATION-->$-gradient estimates for the positive solutions to ${\mathrm{\Delta }}_{p}u+a{u}^q=0$ on $(M,g)$ with Ricci lower bound hold true is equivalent to whether or not the positive solutions to this equation fulfill Harnack inequality, and hence some new Cheng–Yau $\log <!--FUNCTION\; APPLICATION-->$-gradient estimates are established.

Keywords

gradient estimate, Nash–Moser iteration, Liouville type theorem

Mathematical Subject Classification

Primary: 35A01, 35B09, 35R01

Milestones

Received: 30 April 2025

Revised: 24 January 2026

Accepted: 26 February 2026

Published: 23 April 2026