Quasilinear equations with source terms on Carnot groups

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 G of arbitrary step. The quasilinear part involves operators of the p-Laplacian type ∆G, p , 1 < p < ∞. 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.