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

Abstract

Abstract:

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

-Δ_{H, p}u=λg(ξ)|u|^q-2u+f (ξ)|u|^p*-2u, in Hⁿ,

u ∈ D^{1, p}(Hⁿ),

where ξ ∈ Hⁿ, λ ∈ R,1<p<Q=2n+2, n ≥ 1, 1<q<p, p*=Qp/Q-p, g(ξ) and f(ξ) 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.

Key words: Heisenberg group, p-sub-Laplacian, Concentration-compactness principle, Palais-Smale condition, Multiplicity

CLC Number:

35D05

DOU Jing-Bo, GUO Qian-Qiao. A Concentration-Compactness Principle at Infinity on the Heisenberg Group and Multiplicity of Solutions for p-sub-Laplacian Problem Involving Critical Sobolev Exponents[J].Acta mathematica scientia,Series A, 2009, 29(4): 1033-1043.

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

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