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 Riemannian surfaces, we have an estimate of type sup + inf. The second result concern the solutions of prescribed scalar curvature equation on the unit ball of ℝ n with Dirichlet condition. Next, we give an inequality of type (sup κ u) 2s−1 × inf Ω u ≤ c for positive solutions of Δu=Vu 5 on Ω ⊂ R 3, where K is a compact set of Ω and V is s-Hölderian, s∈]-1/2,1]. For the case s=1/2 and Ω = S 3, we prove that, if minΩ u>m>0 (for some particular constant m >0), and the Hölderian constant A of V tends to 0 (in certain meaning), we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of Ω.