## Quasilinear equations with source terms on Carnot groups

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- by Nguyen Cong Phuc and Igor E. Verbitsky PDF
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**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.## References

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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