A priori bounds for positive solutions of subcritical elliptic equations

TL;DR

This paper establishes uniform bounds for positive solutions of subcritical elliptic equations in bounded domains using the moving planes method and Kelvin transform, addressing a longstanding open problem.

Contribution

It introduces a novel approach combining Kelvin transform and moving planes to derive a priori bounds independent of the non-linearity.

Findings

Positive solutions are bounded above by a domain-dependent constant.

The bounds are independent of the specific non-linearity.

The method applies to a class of subcritical elliptic problems.

Abstract

We provide a-priori $L^\infty$ bounds for positive solutions to a class of subcritical elliptic problems in bounded $C^2$ domains. Our arguments rely on the moving planes method applied on the Kelvin transform of solutions. We prove that locally the image through the inversion map of a neighborhood of the boundary contains a convex neighborhood; applying the moving planes method, we prove that the transformed functions have no extremal point in a neighborhood of the boundary of the inverted domain. Retrieving the original solution $u$, the maximum of any positive solution in the domain $\Om,$ is bounded above by a constant multiplied by the maximum on an open subset strongly contained in $\Om.$ The constant and the open subset depend only on geometric properties of $\Om,$ and are independent of the non-linearity and on the solution $u$. Our analysis answers a longstanding open problem.