Heisenberg群上无穷远处的集中列紧原理和具有Sobolev临界指数的p-次Laplace方程多解的存在性

A Concentration-Compactness Principle at Infinity on the Heisenberg Group and Multiplicity of Solutions for p-sub-Laplacian Problem Involving Critical Sobolev Exponents

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方程其中ξ∈H^n,λ∈R,1<P<Q=2n+2,n≥1,1<q<p,P~*=(Qp)/(Q-p),g(ξ),f(ξ)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.

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.