LOWER BOUNDS FOR SUP＋INF AND SUP＊INF AND AN EXTENSION OF CHEN-LIN RESULT IN DIMENSION 3

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 Rn with Dirichlet condition. Next, we give an inequality of type （supK ^u）^2s-1 × infπu ≤ c for positive solutions of △u = V u^5 on Ω belong toR^3, where K is a compact set of Ω and V is s-Holderian, s ∈] - 1/2, 1]. For the case s = 1/2 and Ω = S3, we prove that, if minΩ u 〉 m 〉 0 （for some particular constant m 〉 0）, and the H¨olderian 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.

Lower bounds for sup+inf and sup*inf and an extension of chen-lin result in dimension 3

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 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=Vu⁵ on Ω R³, 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 = S₃, 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.