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

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 $({\sup }_{K}u{)}^{2s-1}\times {\inf }_{\Omega }u\le c$ for positive solutions of $\Delta u=V{u}^5$ on $\Omega \subset {\mathbb{R}}^3$, where $K$ is a compact set of $\Omega$ and $V$ is $s-$ hölderian, $s\in ]-1/2,1]$. For the case $s=1/2$, we prove that if ${\min }_{\Omega }u>m>0$ and the hölderian 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.