Journal of Inequalities and Applications 
Volume 2007 (2007), Article ID 32585, 24 pages
doi:10.1155/2007/32585
Research Article

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

Jingbo Dou, Pengcheng Niu, and Zixia Yuan

 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
Received 22 March 2007; Revised 26 May 2007; Accepted 20 October 2007

Recommended by László Losonczi

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 p-sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.