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

We prove an analogue of Wolff’s inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation $$-{\mathrm{\Delta }}_{p}u=\sigma u^q\text{in}{\mathbb{R}}^n,$$ in the sub-natural growth case $0<q<p-1$, where $\Delta_{p}u = $\text{div}$( |\nabla u|^{p-2} \nabla u )$isthe$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 $u\in L^r({\mathbb{R}}^n)$ ($0<r<\mathrm{\infty }$) 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 $f^{p, q}(\sigma)$definedintermsofcharacteristicfunctionsofdyadiccubesin$$\mathbb{R}$^n$.