Quasilinear equations with source terms on Carnot groups

Phuc, Nguyen; Verbitsky, Igor

doi:10.1090/S0002-9947-2013-05920-X

Quasilinear equations with source terms on Carnot groups
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by Nguyen Cong Phuc and Igor E. Verbitsky PDF

Trans. Amer. Math. Soc. 365 (2013), 6569-6593 Request permission

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 we 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.

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