A Hardy inequality with remainder terms in the Heisenberg group and the weighted eigenvalue problem

Abstract

Based on properties of vector fields, we prove Hardy inequalities with remainderterms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The bestconstants in Hardy inequalities are determined. Then we discuss the existence of solutions forthe 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.