A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

TL;DR

This paper establishes uniform bounds for positive solutions of a class of semi-linear elliptic equations with critical nonlinearity under specific conditions on the coefficient function, providing key a priori estimates for solutions on the unit ball and sphere.

Contribution

It introduces a novel a priori estimate for positive solutions of semi-linear elliptic equations with critical exponent, based on the sub-harmonicity of the coefficient function at critical points.

Findings

Solutions are uniformly bounded under given conditions.

The estimate applies to equations on the sphere.

Conditions involve the sub-harmonicity of the coefficient function.

Abstract

We consider positive solutions of $\Delta u-\mu u+Ku^{\frac{n+2}{n-2}}=0$ on $B_1$ ($n\ge 5$) where $\mu $ and $K>0$ are smooth functions on $B_1$. If $K$ is very sub-harmonic at each critical point of $K$ in $B_{2/3}$ and the maximum of $u$ in $\bar B_{1/3}$ is comparable to its maximum over $\bar B_1$, then all positive solutions are uniformly bounded on $\bar B_{1/3}$. As an application, a priori estimate for solutions of equations defined on $\mathbb S^n$ is derived.