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

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