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PAPER: A class of sub-elliptic equations on the Heisenberg group and related interpolation inequalities

REF. DATE:5 February 2013 CREATED/MODIFIED: 15 January 2016 199 views No Comment

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)

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