A comparison principle for the Lane–Emden equation and applications to geometric estimates. (July 2022)

• Record Type: Journal Article
• Authors: Brasco, Lorenzo; Prinari, Francesca; Zagati, Anna Chiara
• Is Part Of: Nonlinear analysis. Volume 220(2022)
• Publication Date: 2022-07
• Subjects: 35B51 -- 35J92 -- 49R05; Comparison principle -- p-Laplacian -- Lane–Emden equation -- Inradius -- Convex sets; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112847
• Languages: English
• ISSNs: 0362-546X

Abstract

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.