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 \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n, \] 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 $\mathbb{R}^n$.