A comparison principle for the Lane-Emden equation and applications to geometric estimates

TL;DR

This paper establishes a comparison principle for the Lane-Emden equation involving the p-Laplacian, enabling new geometric and analytical estimates for solutions without regularity constraints.

Contribution

It introduces a novel comparison principle for positive solutions to the Lane-Emden equation that applies broadly and facilitates various geometric and uniqueness results.

Findings

Proved a comparison principle for supersolutions and subsolutions.

Derived sharp pointwise estimates for solutions in convex domains.

Established localization and geometric estimates for principal frequencies.

Abstract

We prove a comparison principle for positive supersolutions and subsolutions to the Lane-Emden equation for the $p-$Laplacian, with subhomogeneous power in the right-hand side. The proof uses variational tools and the result applies with no regularity assumptions, both on the set and the functions. We then show that such a comparison principle can be applied to prove: uniqueness of solutions; sharp pointwise estimates for positive solutions in convex sets; localization estimates for maximum points and sharp geometric estimates for generalized principal frequencies in convex sets.