Critical Kirchhoff equations involving the p-sub-Laplacians operators on the Heisenberg group

• Published in: Bulletin of mathematical sciences Vol. 13; no. 2
• Main Authors: Sun, Xueqi, Song, Yueqiang, Liang, Sihua, Zhang, Binlin
• Format: Journal Article
• Language: English
• Published: World Scientific Publishing Company 01.08.2023
• Subjects: variational methods; Heisenberg group; sub-Laplacian; concentration-compactness principles
• ISSN: 1664-3607; 1664-3615
• DOI: 10.1142/S1664360722500060

In this paper, we deal with a class of Kirchhoff-type critical elliptic equations involving the p -sub-Laplacians operators on the Heisenberg group of the form M ( ∥ D H u ∥ p p + ∥ u ∥ p , V p ) [ − Δ H , p u + V ( ξ ) | u | p − 2 u ] = λ f ( ξ , u ) + | u | p ∗ − 2 u , ξ ∈ ℍ n , u ∈ HW V 1 , p ( ℍ n ) , where Δ H , p u : = div H ( | D H u | H p − 2 D H u ) is the p -sub-Laplacian, 1 < p < Q , HW V 1 , p ( ℍ n ) is the horizontal Sobolev space on ℍ n . And Q = 2 n + 2 is the homogeneous dimension of ℍ n , λ is a real parameter, p ∗ = Q p / ( Q − p ) is the critical Sobolev exponent on the Heisenberg group. Under some proper assumptions on the Kirchhoff function M , the potential function V and f , together with the mountain pass theorem and the concentration-compactness principles for classical Sobolev spaces on the Heisenberg group, we prove the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases on the Heisenberg group. The results of this paper extend or complete recent papers and are new in several directions for the non-degenerate and degenerate critical Kirchhoff equations involving the p -Laplacian type operators on the Heisenberg group.