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

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. Contents 1. Introduction 1.1. The Lane-Emden equation 1.2. Main results 1.3. Some comments 1.4. Plan of the paper 2. Preliminaries 2.1. Notation 2.2. Sobolev spaces 2.3. Weak solutions 2.4. Hidden convexity 3. Some properties of the energy functional 4. A comparison principle 5. Applications to geometric estimates 5.1. Solutions of the Lane-Emden equation 5.2. Localization of maximum points 5.3. Generalized principal frequencies Appendix A. Quantified convexity of power functions Appendix B. Some subsolutions Appendix C. Asymptotics of the positive solution in a slab-type sequence References

Uniqueness of the critical point for solutions to some $p$-Laplace equations in the plane

“…For example, the p-torsion function of a square has strictly convex positive super-level sets. More generally, in[9, Theorem 4.1], uniqueness for problem (1.1) in any domain has been proved for…”