Some uniform estimates for scalar curvature type equations

We consider the prescribed scalar curvature equation on an open set $Ω$ of $ℝ^{n}$ , $- Δ u = V u^{(n + 2) ∕ (n - 2)} + u^{n ∕ (n - 2)}$ with $V \in C^{1, α}$ ( $0 < α \leq 1$ ), and we prove the inequality $sup_{K} u \times inf_{Ω} u \leq c$ where $K$ is a compact set of $Ω$ .

In dimension 4, we have an idea on the supremum of the solution of the prescribed scalar curvature if we control the infimum. For this case we suppose the scalar curvature $C^{1, α}$ ( $0 < α \leq 1$ ).

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Keywords

$\sup \times \inf$, nonlinear perturbation, dimension 4, minimal conditions

Mathematical Subject Classification 2010

Primary: 35B44, 35B45, 35B50, 35J60, 53C21

Milestones

Received: 7 November 2017

Revised: 24 October 2018

Accepted: 2 November 2018

Published: 16 September 2019

Authors

Samy Skander Bahoura
Equipe d’Analyse Complexe et Géométrie Universite Pierre et Marie Curie Paris France

Vol. 301, No. 1, 2019

Samy Skander Bahoura

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors