Calculus of Variations and Geometric Measure Theory

L. Brasco - F. Prinari - A. C. Zagati

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

created by zagati on 19 Nov 2021
modified by brasco on 17 Mar 2022


Accepted Paper

Inserted: 19 nov 2021
Last Updated: 17 mar 2022

Journal: Nonlinear Anal.
Pages: 43
Year: 2022

ArXiv: 2111.09603 PDF


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.

Keywords: p-Laplacian, convex sets, Comparison principle, Inradius, Lane-Emden equation