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

数学物理学报 ›› 2009, Vol. 29 ›› Issue (4): 1033-1043.

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

(1.西安财经学院统计学院, 西安 710061, 2.西北工业大学应用数学系, 西安 710072)

收稿日期:2008-04-18 修回日期:2009-05-27 出版日期:2009-08-25 发布日期:2009-08-25
基金资助:
陕西省自然科学基础研究计划(2006A09)和西北工业大学科技创新基金(2008kJ02033)资助

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

(1.School of Statistics, Xi'an Institute of Finance and Economics, Xi'an 710061, 2.Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072)

Received:2008-04-18 Revised:2009-05-27 Online:2009-08-25 Published:2009-08-25
Supported by:
陕西省自然科学基础研究计划(2006A09)和西北工业大学科技创新基金(2008kJ02033)资助

摘要/Abstract

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

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

u ∈ D^{1, p}(Hⁿ),

其中ξ ∈ Hⁿ, λ ∈ R, 1<p<Q=2n+2, n ≥ 1, 1<qp*=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.

关键词: Heisenberg 群, p -次Laplace算子, 集中列紧原理, Palais-Smale条件, 多解

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

中图分类号:

35D05

窦井波, 郭千桥. 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[J]. 数学物理学报, 2009, 29(4): 1033-1043.

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.

参考文献

[1] Lions P L. The concentration-compactness principle in the calculus of variations; the limit case, Part 1,2. Revista Mat Ibaroamericana, 1985, 1(2/3): 145--201, 45--121

[2] Lions P L. The concentration-compactness principle in the calculus of variations, locaily compact case, Part 1,2. Ann Inst H Poincar'e, 1984, 1(1/4): 109--145, 223--283

[3] Bianchi G, Chabrowski J, Szulkin A. On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev
exponent. Nonlinear Analysis, 1995, 25: 41--59

[4] Ben-Naoum A K, Troestler C, Willem M. Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Analysis, 1996, 26: 823--833

[5] Chabrowski J. Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev
exponents. Calc Var Partial Differential Equations, 1995, 3: 493--512

[6] Huang D, Li Y. A concentration-compactness principle at infinity and positive solutions of some quasilinear elliptic equations in
unbounded domains. J Math Anal Appl, 2005, 304: 58--73

[7] Folland G B, Stein E M. Estimaties for the $\bar{\partial}_b$-complex and analysis on the Heisenberg group. Comm Pure Appl Math, 1974, 27: 429--522

[8] Niu P, Zhang H, Wang Y. Hardy type and Rellich type inequalities on the Heisenberg group. Proc Amer Math Soc, 2001, 129(12): 3623--3630

[9] Garofalo N, Lanconelli E. Existence and nonexistence results for semilinear equations on the Heisenberg group. Indiana Uni Math J, 1992, 41(1): 71--98

[10] Wang W. Positive solution of a subelliptic nonlinear equation on the heisenberg group. Canadian Math Bulletin, 2001, 43(3): 346--354

[11] Garofalo N, Vassilev D. Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups. Math Ann, 2000, 318: 453--516

[12] Huang Y S. On multiple solutions of quasilinear equations involving the critical sobolev exponent. J Math Anal Appl, 1999, 231: 142--160

[13] Yang J. Positive solutions of quasilinear elliptic obstacle problems with critical exponents. Nonlinear Analysis, 1995, 25: 1283--1306

[14] Dr'abek P, Huang Y X. Multiplicity of positive solutions for some quasilinear elliptic equation in ${\Bbb R}^N$ with critical Sobolev exponent. J Diff Eqs, 1997, 140: 106--132

[15] Garofalo N, Nhieu D M. Isoperimetric and Sobolev inequalities for Carnot-Carth'eodory spaces and the existence of minimal surfaces. Comm Pure Appl Math, 1996, 49: 1081--1144

[16] Clark D C. A variant of Ljusternik-Schnirelman theory. Indiana Univ Math J, 1972, 22: 65--74

[17] Struwe M. Variational Methods and Their Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Second Edition.
Berlin: Springer-Verlag, 2000