A comparison principle for the Lane–Emden equation and applications to geometric estimates. (July 2022) - Brasco, Lorenzo; Prinari, Francesca; Zagati, Anna Chiara
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.