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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quasilinear equations with source terms on Carnot groups


Authors: Nguyen Cong Phuc and Igor E. Verbitsky
Journal: Trans. Amer. Math. Soc. 365 (2013), 6569-6593
MSC (2010): Primary 35H20; Secondary 35A01, 20F18
Published electronically: June 3, 2013
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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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Additional Information

Nguyen Cong Phuc
Affiliation: Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, Louisiana 70803
Email: pcnguyen@math.lsu.edu

Igor E. Verbitsky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: verbitskyi@missouri.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05920-X
PII: S 0002-9947(2013)05920-X
Received by editor(s): December 18, 2011
Received by editor(s) in revised form: July 3, 2012
Published electronically: June 3, 2013
Additional Notes: The first author was supported in part by NSF Grant DMS-0901083
The second author was supported in part by NSF Grant DMS-0901550
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.