Uniform a priori estimates for positive solutions of the Lane-Emden equation in the plane

TL;DR

This paper establishes uniform bounds for positive solutions of the Lane-Emden equation in two-dimensional bounded domains as the exponent grows large, contributing to the understanding of solution behavior in nonlinear PDEs.

Contribution

It provides the first uniform a priori estimates for positive solutions of the Lane-Emden equation in the plane for large exponents.

Findings

Positive solutions are uniformly bounded for large exponents.

The bounds depend only on the domain and not on the exponent.

The result advances the theory of nonlinear elliptic equations in two dimensions.

Abstract

We prove that positive solutions of the Lane-Emden equation in a two-dimensional smooth bounded domain are uniformly bounded for all large exponents.