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Liouville theorems and gradient estimates of a nonlinear elliptic equation for the V-Laplacian

Yike Jia

TL;DR

The paper investigates gradient estimates and Liouville-type results for positive solutions of the nonlinear elliptic equation $Δ_{V}u^{m}+μ(x)u+p(x)u^{α}=0$ on smooth metric measure spaces with $k$-Bakry-Émery curvature bounded below by $-(k-1)K$. By introducing $v=\frac{m}{m-1}u^{m-1}$ and the weighted gradient quantity $ω=|\nabla v|^{2}/v$, the authors construct the cutoff function framework $G=\varphi ω$ and apply the weighted Bochner formula to derive differential inequalities, handling the regimes $α\ge1$ and $α<1$ to obtain explicit gradient bounds. These estimates yield Liouville-type theorems and Harnack inequalities, and extend known results (e.g., Wang) to the $Δ_V$-Laplacian setting, including special cases where $V=0$ or $p=0$. The results demonstrate how geometry, via $Ric_V^k$, interacts with nonlinear elliptic PDEs on weighted manifolds, providing rigidity when curvature is nonnegative and coefficients are nonnegative or constant. The work broadens the toolkit for nonlinear elliptic analysis on metric measure spaces and informs further studies on gradient estimates and rigidity phenomena.

Abstract

In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$Δ_{V}u^{m}+μ(x)u+p(x)u^α=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-Émery curvature is bounded from below by $-(k-1)K$ with $K \geq 0$. Additionally, we obtain related Liouville theorems and Harnack inequalities. We partially extend conclusions of Wang, when $V=0$, $μ=0$ the equation becomes $Δu^{m}+p(x)u^α=0$. And $V=f$, $μ=c, p=0 $, the equation becomes $Δ_{f}u^{m}+cu=0 $.

Liouville theorems and gradient estimates of a nonlinear elliptic equation for the V-Laplacian

TL;DR

The paper investigates gradient estimates and Liouville-type results for positive solutions of the nonlinear elliptic equation on smooth metric measure spaces with -Bakry-Émery curvature bounded below by . By introducing and the weighted gradient quantity , the authors construct the cutoff function framework and apply the weighted Bochner formula to derive differential inequalities, handling the regimes and to obtain explicit gradient bounds. These estimates yield Liouville-type theorems and Harnack inequalities, and extend known results (e.g., Wang) to the -Laplacian setting, including special cases where or . The results demonstrate how geometry, via , interacts with nonlinear elliptic PDEs on weighted manifolds, providing rigidity when curvature is nonnegative and coefficients are nonnegative or constant. The work broadens the toolkit for nonlinear elliptic analysis on metric measure spaces and informs further studies on gradient estimates and rigidity phenomena.

Abstract

In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation on any smooth metric measure space whose -Bakry-Émery curvature is bounded from below by with . Additionally, we obtain related Liouville theorems and Harnack inequalities. We partially extend conclusions of Wang, when , the equation becomes . And , , the equation becomes .
Paper Structure (3 sections, 6 theorems, 70 equations)

This paper contains 3 sections, 6 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of main theorems.

Key Result

Theorem 1.1

Let $(M^{n},g)$ be a complete Rimannian mainfold without boundary. Assume that $B_{2R}$ is a geodesic ball of radius $2R$ centered at $q\in M$ and $Ric_V^k(B_{2R}) \geq -K$ with $K \geq 0$. Additionally, suppose there are two positive numbers $\delta_{1},\delta_{2}$ such that $|p(x,t)|\leq\delta_{1} $( 2)$ Assume that $\alpha < 1$ , $\beta > 0$ then Where $C_{4}$ is a constant depending on $k$,

Theorems & Definitions (6)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1