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A class of sub-elliptic equations on the Heisenberg group and related interpolation inequalities (with J. Chen) in Advances in Harmonic Analysis and Operator Theory, Operator Theory: Advances and Applications 229, 123–137, (2013).


ABSTRACT: We firstly prove the existence of least energy solutions to a class of sub-elliptic equations on the Heisenberg group \mathbb{H}^N of the form

    \[-\Delta_H u + u = |u|^{p-2}u\quad u\in S^{1,2}_0(\mathbb{H}^N),\]

where S^{1,2}_0(\mathbb{H}^N) is the closure of C^{\infty}_0(\mathbb{H}^N) under the norm \|u\|=(\int_{\mathbb{H}^N}|\nabla_H u|^2+|u|^2)^{\frac{1}{2}}. Then we use this least energy solution to give a sharp estimate to the smallest positive constant in the Gagliardo-Nirenberg inequality on the Heisenberg group, 2 < p < 2+\frac{2}{N},

    \[\int_{\mathbb{H}^N}|u|^p\,d\xi \leq C\left(\int_{\mathbb{H}^N}|\nabla_H u|^2\,d\xi\right)^{\frac{Q(p-2)}{4}}\left(\int_{\mathbb{H}^N}|u|^2\,d\xi\right)^{\frac{2p-Q(p-2)}{4}}.\]

We also point out some extensions to the quasilinear sub-elliptic case.


Download: Preprint version (RIA |  download) / Publisher version (doi:10.1007/978-3-0348-0516-2)
Extra info: (P085/2013-02-05)

REF. DATE: 5 February 2013 CREATED/MODIFIED: 15 January 2016 VIEW POST 204 views No Comment

Existence of solutions of a class of nonlinear singular equations in Lorentz spaces (with L. Huang, K. Murillo), Advances in Harmonic Analysis and Operator Theory (the Stefan Samko anniversary volume, Integral Differential Operators and their Applications; Eds. Alexandre Almeida, Luis Castro, Frank-Olme Speck), pp.14, accepted in 2012.

REF. DATE: 22 January 2012 CREATED/MODIFIED: 5 September 2012 VIEW POST 259 views No Comment

Disseminating and preserving the mathematical knowledge (with J.F. Rodrigues), in: Communicating Mathematics in the Digital Era, J. Borwein et al. (eds), 3-21, A. K. Peters Ltd, 2008.
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REF. DATE: 8 September 2008 CREATED/MODIFIED: 14 January 2016 VIEW POST 215 views No Comment

The IntBook concept as an adaptative web environment (with A. Breda, T. Parreira), in: Communicating Mathematics in the Digital Era, J. Borwein et al. (eds), 253-268, A. K. Peters Ltd, 2008.
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REF. DATE: 8 September 2008 CREATED/MODIFIED: 19 January 2012 VIEW POST 279 views No Comment

On computation of the logarithm of the Chen-Fliess series for nonlinear systems, in: Nonlinear and Adaptive Control, Zinober et al. (eds), Lect. Notes Control Inf. Sci. 281, 317-326, Springer-Verlag, 2003.
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REF. DATE: 8 September 2003 CREATED/MODIFIED: 18 January 2012 VIEW POST 189 views No Comment