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Nash-Moser iteration approach to the logarithmic gradient estimates and Liouville Properties of quasilinear elliptic equations on manifolds

Jie He, Jingchen Hu, Youde Wang

Abstract

In this paper, we provide a new routine to employ the Nash-Moser iteration technique to analyze the local and global properties of positive solutions to the equation $$Δ_pv + a|\nabla v|^qv^r =0$$ on a complete Riemannian manifold with Ricci curvature bounded from below, where $p>1$, $q$, $r$ and $a$ are some real constants. Assuming certain conditions on $a,\, p,\, q$ and $r$, we can derive universal and succinct Cheng-Yau type logarithmic gradient estimates for such solutions. In particular, we give the obvious expressions of constants in the logarithmic gradient estimate for entire solutions to the above equation (see \thmref{t10}). The gradient estimates enable us to obtain some Liouville-type theorems, Harnack inequalities and some local estimates near singularities for positive solutions. Some of our results are new even in the case the domain is an Euclidean space and $p=2$.

Nash-Moser iteration approach to the logarithmic gradient estimates and Liouville Properties of quasilinear elliptic equations on manifolds

Abstract

In this paper, we provide a new routine to employ the Nash-Moser iteration technique to analyze the local and global properties of positive solutions to the equation on a complete Riemannian manifold with Ricci curvature bounded from below, where , , and are some real constants. Assuming certain conditions on and , we can derive universal and succinct Cheng-Yau type logarithmic gradient estimates for such solutions. In particular, we give the obvious expressions of constants in the logarithmic gradient estimate for entire solutions to the above equation (see \thmref{t10}). The gradient estimates enable us to obtain some Liouville-type theorems, Harnack inequalities and some local estimates near singularities for positive solutions. Some of our results are new even in the case the domain is an Euclidean space and .
Paper Structure (18 sections, 30 theorems, 253 equations, 2 figures)

This paper contains 18 sections, 30 theorems, 253 equations, 2 figures.

Table of Contents

  1. Introduction
  2. History and motivation
  3. Main results and some remarks
  4. Applications to singularity analysis in $\mathbb R^n$.
  5. Global gradient estimates
  6. preliminary
  7. Proof of Theorem \ref{['t4']}
  8. Gradient estimates and applications
  9. Integral inequality
  10. bound of gradient in a geodesic ball with radius $3R/4$
  11. Moser iteration
  12. Proof of Theorem \ref{['thm1.4']}
  13. Proof of Theorem \ref{['thm1.5']}
  14. The case $\dim(M)=2$.
  15. Some applications for the equations on $\Omega\subset\mathbb R^n$.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be complete non-compact manifold with non-negative Ricci curvature. Suppose that $1<p<n$ and $r>0,\, q\geq0,\, q+r>p-1$, and $a>0$. If then the problem admits no solution $v\in C^1(M)$.

Figures (2)

  • Figure 1: The region of $(q,r)$ where Liouville result holds when $n=4$ and $p=3$.
  • Figure 2: The green region is the $(q,r)$ where $\frac{|\nabla v|}{v}$ is bounded by $\frac{(n-1)\sqrt{\kappa}}{p-1}$;The yellow region is the $(q,r)$ where $\frac{|\nabla v|}{v}$ is bounded by $\frac{2\sqrt{\kappa}}{(p-1)\sqrt{\left(\frac{q+r}{p-1}-1\right)\left(\frac{n+3}{n-1}-\frac{q+r}{p-1}\right)}}.$

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Corollary 1.3
  • Remark 2
  • Theorem 1.4
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1.5
  • ...and 43 more