Open Access

A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem

Journal of Inequalities and Applications20072007:032585

DOI: 10.1155/2007/32585

Received: 22 March 2007

Accepted: 20 October 2007

Published: 31 December 2007

Abstract

Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the https://static-content.springer.com/image/art%3A10.1155%2F2007%2F32585/MediaObjects/13660_2007_Article_1694_IEq1_HTML.gif -sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.

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Authors’ Affiliations

(1)
Department of Applied Mathematics, Northwestern Polytechnical University

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Copyright

© Jingbo Dou et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.