Liouville-type theorems for unbounded solutions of elliptic equations in half-spaces

TL;DR

This paper establishes Liouville-type theorems for unbounded solutions of the Lane-Emden equation in half-spaces, showing that solutions with certain growth rates do not exist, extending previous results beyond bounded solutions.

Contribution

The paper proves nonexistence of certain unbounded solutions to the Lane-Emden equation in half-spaces, using new local gradient bounds, thus generalizing prior bounded solution results.

Findings

No positive solutions with specific growth in half-spaces

Established local bounds for the logarithmic gradient

Extended nonexistence results to unbounded solutions

Abstract

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words, we rule out {\it type~I grow-up} solutions. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. Instrumental in the proof are local pointwise bounds for the logarithmic gradient of the solution and its normal derivative, which we also establish.