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On the universal local and global properties of positive solutions to $Δ_pv+b|\nabla v|^q+cv^r=0$ on complete Riemannian manifolds

Jie He, Youde Wang

Abstract

In this paper we study the positive solutions to a nonlinear elliptic equation $$Δ_pv+b|\nabla v|^q+cv^r =0$$ defined on a complete Riemannian manifold $(M,g)$ with Ricci curvature bounded from below, where $p>1$, $q,\, r, \, b$ and $c$ are some real constants. If $p>1$ is given and $bc\geq 0$, we provide a new routine to give some regions of $(q, r)$ such that the Cheng-Yau's logarithmic gradient estimates hold true exactly on such given regions. In particular, we derive the upper bounds of the constants $c(n, p, q, r)$ in the Cheng-Yau's gradient estimates for the entire solutions to the above equation. As applications, we reveal some universal local and global properties of positive solutions to the equation. On the other hand, we extend some results due to \cite{MR1879326} to the case the domain of the equation is a complete manifold and obtain wider ranges of $(q,r)$ for Liouville properties.

On the universal local and global properties of positive solutions to $Δ_pv+b|\nabla v|^q+cv^r=0$ on complete Riemannian manifolds

Abstract

In this paper we study the positive solutions to a nonlinear elliptic equation defined on a complete Riemannian manifold with Ricci curvature bounded from below, where , and are some real constants. If is given and , we provide a new routine to give some regions of such that the Cheng-Yau's logarithmic gradient estimates hold true exactly on such given regions. In particular, we derive the upper bounds of the constants in the Cheng-Yau's gradient estimates for the entire solutions to the above equation. As applications, we reveal some universal local and global properties of positive solutions to the equation. On the other hand, we extend some results due to \cite{MR1879326} to the case the domain of the equation is a complete manifold and obtain wider ranges of for Liouville properties.
Paper Structure (17 sections, 34 theorems, 248 equations, 7 figures)

This paper contains 17 sections, 34 theorems, 248 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Local gradient estimate
  3. Applications to singularity analysis on $\mathbb R^n$
  4. Liouville Theorem
  5. Global gradient estimate
  6. Preliminaries
  7. Gradient estimate for the solutions to (\ref{['equ0']})
  8. Integral inequality on the solutions to (\ref{['equ0']}): the case $\dim(M)>2$
  9. bound of gradient in a ball with radius $3R/4$: the case $\dim(M)>2$
  10. Moser iteration
  11. The case $\dim(M)=2$
  12. Proof of Theorem \ref{['t3']}-Theorem \ref{['t4']}
  13. Singularity analysis on $\mathbb R^n$
  14. Liouville Thoerems
  15. Global gradient estimate
  16. ...and 2 more sections

Key Result

Theorem 1

(Filippucci, Sun and Zheng filippucci2022priori) Let $v$ be a positive solution to the following equation If $(q, r)$ satisfies then there exist positive constants $a$ and $c(n,p,q,r)$ such that

Figures (7)

  • Figure 1: When $b>0, c>0$, the region of $(\frac{q}{p-1}, \frac{r}{p-1})$ in $W_1, W_2, W_3, W_4$.
  • Figure 2: When $b<0, c<0$, the region of $(\frac{q}{p-1}, \frac{r}{p-1})$ in $W_1, W_2, W_3, W_4$.
  • Figure 3: The shadow region with north east lines represents the $(\frac{q}{p-1}, \frac{r}{p-1})$ in filippucci2022priori and MR4150912; the region filled with green color is our region of $(\frac{q}{p-1}, \frac{r}{p-1})$ for gradient estimate.
  • Figure 4: The region of $(\frac{q}{p-1}, \frac{r}{p-1})$ in Theorem \ref{['t6']}.
  • Figure 5: The region filled with green color represents the $(\frac{q}{p-1}, \frac{r}{p-1})$ in Corollory \ref{['c14']}; the region filled with north west lines represents the $(\frac{q}{p-1}, \frac{r}{p-1})$ in Proposition \ref{['p5']}(shadow).
  • ...and 2 more figures

Theorems & Definitions (61)

  • Theorem
  • Theorem 1.1
  • Corollary 1.2
  • Remark 1
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 2
  • Corollary 1.6
  • Remark 3
  • ...and 51 more