A Liouville theorem for the Lane-Emden system in the half-space

TL;DR

This paper establishes a Liouville theorem showing the nonexistence of positive bounded solutions for the Lane-Emden system in a half-space, extending previous results to unbounded solutions without restrictions on nonlinear powers.

Contribution

It proves a new nonexistence result for positive solutions of the Lane-Emden system in a half-space without restrictions on solution boundedness or nonlinear exponents.

Findings

No positive classical solutions bounded on finite strips exist in the half-space.

The result extends previous nonexistence theorems to unbounded solutions.

The theorem applies to the Dirichlet problem for the Lane-Emden system.

Abstract

We prove that the Dirichlet problem for the Lane-Emden system in a half-space has no positive classical solution that is bounded on finite strips. Such a nonexistence result was previously available only for bounded solutions or under a restriction on the powers in the nonlinearities.