Lower bounds for sup + inf and sup * inf and an Extension of Chen-Lin result in dimension 3

TL;DR

This paper establishes new Harnack type inequalities for solutions to elliptic equations on Riemannian surfaces and in three dimensions, extending previous results and providing bounds under specific conditions.

Contribution

It introduces novel inequalities for elliptic PDE solutions on surfaces and in three dimensions, extending Chen-Lin results and analyzing solutions with Hölder continuous coefficients.

Findings

Derived a $ ext{sup}+ ext{inf}$ estimate on Riemannian surfaces.

Established an inequality involving $ ext{sup}_K u$ and $ ext{inf}_ abla u$ for $ riangle u=Vu^5$ in $ extbf{R}^3$.

Proved uniform boundedness of solutions under small Hölderian constants for $V$.

Abstract

We give two results about Harnack type inequalities. First, on compact smooth Riemannian surface without boundary, we have an estimate of the type $\sup +\inf$. The second result concerns the solutions of prescribed scalar curvature equation on the unit ball of ${\mathbb R}^n$ with Dirichlet condition. Next, we give an inequality of the type $(\sup_K u)^{2s-1} \times \inf_{\Omega} u \leq c$ for positive solutions of $\Delta u=Vu^5$ on $\Omega \subset {\mathbb R}^3$, where $K$ is a compact set of $\Omega$ and $V$ is $s-$ h\"olderian, $s\in ]-1/2,1]$. For the case $s=1/2$, we prove that if $\min_{\Omega} u>m>0$ and the h\"olderian constant $A$ of $V$ is small enough (in certain meaning), we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of $\Omega$. ----- Nous donnons quelques estimations des solutions d'equations elliptiques sur les surfaces de Riemann et sur des ouverts en dimension n> 2. Nous traitons le cas holderien pour l'equation de la courbure scalaire prescrite en dimension 3.