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Global and local properties of solutions of elliptic equations with a nonlinear term involving the product of the function and its gradient

Zhihao Lu

Abstract

We study the global and local properties of positive solutions to the quasi-linear elliptic equation: \d u+|\nabla u|^q u^p=0,\quad x\in Ø\subset \mathbb{R}^n,\nonumber where $q\ge 0$ and $p\in\mathbb{R}$. Our contributions are twofold: 1. Based on an optimal and new identity for the modulus squared of the logarithmic gradient, we establish optimal and improved Liouville theorems for global positive solutions, and generalize these findings to the framework of Riemannian manifolds. 2. Based on a newly discovered mutual control relationship of two nonlinear iterms, for all index pairs \( (p, q) \) where the Liouville theorem holds, we derive several optimal gradient estimates for local positive solutions. As a direct corollary, we obtain the corresponding Harnack inequality. These results strengthen the related conclusions in Bidaut-Véron--García-Huidobro--Véron \cite {BGV} from both global and local perspectives.

Global and local properties of solutions of elliptic equations with a nonlinear term involving the product of the function and its gradient

Abstract

We study the global and local properties of positive solutions to the quasi-linear elliptic equation: \d u+|\nabla u|^q u^p=0,\quad x\in Ø\subset \mathbb{R}^n,\nonumber where and . Our contributions are twofold: 1. Based on an optimal and new identity for the modulus squared of the logarithmic gradient, we establish optimal and improved Liouville theorems for global positive solutions, and generalize these findings to the framework of Riemannian manifolds. 2. Based on a newly discovered mutual control relationship of two nonlinear iterms, for all index pairs \( (p, q) \) where the Liouville theorem holds, we derive several optimal gradient estimates for local positive solutions. As a direct corollary, we obtain the corresponding Harnack inequality. These results strengthen the related conclusions in Bidaut-Véron--García-Huidobro--Véron \cite {BGV} from both global and local perspectives.
Paper Structure (12 sections, 40 theorems, 385 equations, 15 figures)

This paper contains 12 sections, 40 theorems, 385 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. New contributions and ideas
  5. Organization of the paper
  6. The framework of Liouville theorems
  7. A fundamental lemma
  8. Proof of Theorem \ref{['t10']} and \ref{['sc']}
  9. Equations of the auxiliary functions
  10. Proofs of Theorem \ref{['2d']}, \ref{['t27']} and \ref{['t28']}
  11. Mutual control lemmas
  12. Local estimates from Liouville theorems

Key Result

Theorem 1.1

Let $u$ be a nonnegative classical solution of pqle on $\mathbb{R}^n$ with $q\ge 0$ and $p\in \mathbb{R}$. Let $\mathscr{D}(n)$ be defined by NL. If any of the following conditions holds: (1) $n=2$; (2) $q\ge \frac{5}{3}$; (3) $q\in[0,1-\frac{1}{\sqrt{n-1}}]$ and $(p,q)\in \mathbb{R}^2_{+}\setminus

Figures (15)

  • Figure 1: Some separated curves in previous progresses ($n=6$)
  • Figure 2: present Liouville domain of equation \ref{['pqle']} ($n=6$)
  • Figure 3: Liouville domain of equation \ref{['pqle']} with boundedness assumption ($n=6$)
  • Figure 4: Upper bound of $z$ by $\mathbb{H}(p, q)<0$
  • Figure :
  • ...and 10 more figures

Theorems & Definitions (69)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4: Admissible set
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 59 more