#### Vol. 301, No. 1, 2019

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Some uniform estimates for scalar curvature type equations

### Samy Skander Bahoura

Vol. 301 (2019), No. 1, 55–65
##### 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 $\underset{K}{sup}u×\underset{\Omega }{inf}u\le c$ 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