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 \[ -Δ_{p} u = σu^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the sub-natural growth case $0<q<p-1$, where ${\Delta }_{p}u=\text{div}(|\nabla u{|}^{p-2}\nabla u)$ is the $p$-Laplacian, and $\sigma$ is a nonnegative measurable function (or measure) on ${\mathbb{R}}^n$. As an application, we give a necessary and sufficient condition for the existence of a positive solution $u\in L^r({\mathbb{R}}^n)$ ($0<r<\infty$) 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 $f^{p,q}(\sigma )$ defined in terms of characteristic functions of dyadic cubes in ${\mathbb{R}}^n$.

arXiv comment25 pages