Wolff's inequality for intrinsic nonlinear potentials and quasilinear elliptic equations. (May 2020)

• Record Type: Journal Article
• Authors: Verbitsky, Igor E.
• Is Part Of: Nonlinear analysis. Volume 194(2020)
• Publication Date: 2020-05
• Subjects: primary 35J92 42B37 -- secondary 35J20; Nonlinear potentials -- Wolff's inequality -- p-Laplacian; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.04.015
• Languages: English
• ISSNs: 0362-546X

Abstract

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 = σ u q in R n, in the case 0 < q < p − 1 . Here Δ p u = 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 ∈ L r ( R n ) ( 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 f p, q ( σ ) defined in terms of characteristic functions of dyadic cubes in R n .