Wolff’s inequality for intrinsic nonlinear potentials and quasilinear elliptic equations

Abstract

We prove an analogue of Wolff’s inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation −Δpu=σuqinRn,in the case 0<q<p−1. Here Δpu=div(|∇u|p−2∇u) is the p-Laplacian, and σ is a nonnegative measurable function (or measure). As an application, we give a necessary and sufficient condition for the existence of a positive solution u∈Lr(Rn) (0<r<∞) to this problem, which was open even in the case p=2.Our version of Wolff’s inequality for intrinsic nonlinear potentials relies on a new characterization of discrete Littlewood–Paley spaces fp,q(σ) defined in terms of characteristic functions of dyadic cubes in Rn.