中文摘要：

通过建立Heisenberg群上无穷远处的集中列紧原理，研究了如下P-次Laplace方程 -△H，pu=λg（ξ）｜u｜^q-2u＋f（ξ）｜u｜p^＊-2u,在H^n上， u∈D^1,p（H^n）， 其中ξ∈H^n，λ∈R，1〈P〈Q=2n＋2，n≥1，1〈q〈p，P＊=Qp/q-p，g（ξ），f（∈）是可以变号和满足一定条件的函数．在适当条件下利用集中列紧原理证明在某个水平处的Palais-Smale条件，从而结合变分原理得到方程存在m-j对解，其中m〉J，且m，J为整数．

英文摘要：

The main results of this paper establish the concentration-compactness principle at infinity on the Heisenberg group. The authors consider the p-sub-Laplacian problem involving critical Sobolev exponents -△H,pu=λg（ξ）｜u｜^q-2u＋f（ξ）｜u｜p^＊-2u, in H^n, u∈D^1,p（H^n）, where ξ∈H^n,λ∈R,1〈P〈Q=2n＋2,n≥1,1〈q〈p,P＊=Qp/q-p,g（ξ） and f（∈） change sign and satisfy some suitable conditions. Under certain assumptions, they show the existence of m - j pairs of nontrivial solutions via variational method, where m 〉 j, both m and j are integers. The concentration-compactness principle allows to prove the Palais-Smale condition is satisfied below a certain level.