On a global estimate and a Stampacchia-type maximum principle for Lane-Emden systems

TL;DR

This paper proves a global boundedness result for Lane-Emden systems with general elliptic operators and positive exponents, and shows limitations in applying the classical Stampacchia maximum principle to these systems.

Contribution

It introduces a new global boundedness theorem for Lane-Emden systems with general operators and highlights the non-reducibility to classical maximum principles.

Findings

Established a global boundedness result for Lane-Emden systems.

Identified limitations of classical Stampacchia maximum principle for these systems.

Demonstrated results for systems involving Laplacian and other divergence form operators.

Abstract

We establish a global boundedness result for Lane-Emden systems involving general second-order elliptic operators in divergence form and arbitrary positive exponents whose product equals one. Furthermore, we observe that, for this class of systems -- and for certain operators in divergence form, including the case when both operators are the Laplacian -- it is not possible to recover the classical Stampacchia maximum principle as a particular case corresponding to single equations.