Lions-type compactness and Rubik actions on the Heisenberg group

• DOI: 10.1007/s00526-012-0543-y
• Publication Date: 2012-07-06T21:47:20Z
• Authors: Zoltán M. Balogh; Alexandru Kristály
• Keywords: Heisenberg group; Multiplicity (mathematics)

In this paper we prove a Lions-type compactness embedding result for symmetric unbounded domains of the Heisenberg group. The natural group action on the Heisenberg group \({\mathbb{H}^n=\mathbb{C}^n \times \mathbb{R}}\) is provided by the unitary group U(n) × {1} and its appropriate subgroups, which will be used to construct subspaces with specific symmetry and compactness properties in the Folland-Stein’s horizontal Sobolev space \({HW_0^{1,2}(\mathbb{H}^n)}\). As an application, we study the multiplicity of solutions for a singular subelliptic problem by exploiting a technique of solving the Rubik-cube applied to subgroups of U(n) × {1}. In our approach we employ concentration compactness, group-theoretical arguments, and variational methods.