Liouville-type theorems for a class of semilinear elliptic equations with nonlinear gradient terms on smooth metric measure spaces

Fanqi Zeng; Cheng Jin; Fanqi Zeng; Cheng Jin

doi:10.3934/cam.2026027

Communications in Analysis and Mechanics

2026, Volume 18, Issue 3: 649-668. doi: 10.3934/cam.2026027

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Research article

Liouville-type theorems for a class of semilinear elliptic equations with nonlinear gradient terms on smooth metric measure spaces

Fanqi Zeng ,
Cheng Jin ^,

School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

Received: 31 August 2025 Revised: 06 April 2026 Accepted: 03 July 2026 Published: 14 August 2026
58J05, 35B45

In this paper, we consider the following $ f $-Laplacian equation with a gradient term and a nonlinear term
$ \begin{equation*} \Delta_{f} u+W\left(u\right)\left|\nabla u\right|^2+F\left(u\right) = 0 \end{equation*} $
on smooth metric measure spaces with nonnegative $ m $-Bakry-Émery Ricci curvature, where $ W(t) $ is a continuous function for $ t > 0 $ and $ F(t) $ is a differentiable function in $ (0, \infty) $. Under certain assumptions on $ W $ and $ F $, by using the Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method, we derive some Liouville-type theorems for positive solutions to the above equation. Our results extend those of Lu (2025) and McCoy (2007).
- nonlinear elliptic equation,
- Nash-Moser iteration,
- Liouville theorem
Citation: Fanqi Zeng, Cheng Jin. Liouville-type theorems for a class of semilinear elliptic equations with nonlinear gradient terms on smooth metric measure spaces[J]. Communications in Analysis and Mechanics, 2026, 18(3): 649-668. doi: 10.3934/cam.2026027

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Abstract

In this paper, we consider the following $ f $-Laplacian equation with a gradient term and a nonlinear term

$ \begin{equation*} \Delta_{f} u+W\left(u\right)\left|\nabla u\right|^2+F\left(u\right) = 0 \end{equation*} $

on smooth metric measure spaces with nonnegative $ m $-Bakry-Émery Ricci curvature, where $ W(t) $ is a continuous function for $ t > 0 $ and $ F(t) $ is a differentiable function in $ (0, \infty) $. Under certain assumptions on $ W $ and $ F $, by using the Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method, we derive some Liouville-type theorems for positive solutions to the above equation. Our results extend those of Lu (2025) and McCoy (2007).

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