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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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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
  • 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
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 6569-6593
  • MSC (2010): Primary 35H20; Secondary 35A01, 20F18
  • DOI: https://doi.org/10.1090/S0002-9947-2013-05920-X
  • MathSciNet review: 3105763