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

TL;DR

This paper establishes uniform bounds for positive solutions of the Lane-Emden system in two-dimensional domains when the exponents are comparable, ensuring bounded energy, and highlights the importance of exponent relations.

Contribution

It provides uniform a priori estimates for solutions of the Lane-Emden system in 2D, showing boundedness when exponents are comparable, a novel result in this context.

Findings

Solutions are bounded independently of exponents when they are comparable.

Energy of solutions remains uniformly bounded under these conditions.

Boundedness may fail if exponents are not comparable.

Abstract

We prove that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded. In addition, the boundedness may fail if the exponents are not comparable.