Quasilinear equations with source terms on Carnot groups

TL;DR

This paper establishes necessary and sufficient conditions for solutions to quasilinear Lane-Emden equations with measure data on Carnot groups, extending known results to arbitrary step groups and general p-Laplacian operators.

Contribution

It provides new a priori estimates using nonlinear Wolff potentials and characterizes removable singularities and Liouville theorems for these equations on Carnot groups.

Findings

Characterization of solution existence conditions

Complete description of removable singularities

Liouville theorem for supersolutions

Abstract

In this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of Lane--Emden type with measure data on a Carnot group $\mathbb G$ of arbitrary step. The quasilinear part involves operators of the $p$-Laplacian type $\Delta_{\mathbb G,\,p}\,$, $1<p<\infty$. These results are based on new a priori estimates of solutions in terms of nonlinear potentials of Th. Wolff's type. As a consequence, we characterize completely removable singularities, and prove a Liouville type theorem for supersolutions of quasilinear equations with source terms which has been known only for equations involving the sub-Laplacian ($p=2$) on the Heisenberg group.