Some uniform estimates for scalar curvature type equations

Samy Skander Bahoura

Vol. 301 (2019), No. 1, 55–65

DOI: 10.2140/pjm.2019.301.55

Abstract

We consider the prescribed scalar curvature equation on an open set $\Omega$ of ${ℝ}^n$, $-\Delta u=V{u}^{\left(n+2\right)∕\left(n-2\right)}+u^{n∕\left(n-2\right)}$ with $V\in C^{1,\alpha }$ ($0<\alpha \le 1$), and we prove the inequality $\underset{K}{sup}u\times \underset{\Omega }{inf}u\le c$ where $K$ is a compact set of $\Omega$.

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 $C^{1,\alpha }$ ($0<\alpha \le 1$).

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