数学物理学报(英文版) ›› 2004, Vol. 24 ›› Issue (3): 395-402.

• 论文 • 上一篇    下一篇

EXISTENCE OF INFINITELY MANY SOLUTIONS
FOR ELLIPTIC PROBLEMS WITH CRITICAL
EXPONENT

傅红卓,沈尧天   

  • 出版日期:2004-07-20 发布日期:2004-07-20
  • 基金资助:

    Supported by NSFC(10171032); NSF of Guangdong
    Proviance (011606)

EXISTENCE OF INFINITELY MANY SOLUTIONS
FOR ELLIPTIC PROBLEMS WITH CRITICAL
EXPONENT

 FU Gong-Zhuo, CHEN Yao-Tian   

  1. 1.Department of Mathematics, South China University of Technology, Guangzhou 510640, China
    2.Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
  • Online:2004-07-20 Published:2004-07-20
  • Supported by:

    Supported by NSFC(10171032); NSF of Guangdong
    Proviance (011606)

摘要:

This paper is concerned with the following nonlinear Dirichlet problem:
(
−△pu = |u|p∗
−2u + f(x, u) x ∈
,
u = 0 x ∈ @
,
where △pu = div(|∇u|p−2∇u) is the p-Laplacian of u,
 is a bounded domain in Rn(n ≥ 3),
1 < p < n, p = pn
n−p is the critical exponent for the Sobolev imbedding,  > 0 and f(x, u)
satisfies some conditions. It reaches the conclusion that this problem has infinitely many
solutions. Some results as p = 2 or f(x, u) = |u|q−2u, where 1 < q < p, are generalized.

Abstract:

This paper is concerned with the following nonlinear Dirichlet problem:
(
−△pu = |u|p∗
−2u + f(x, u) x ∈
,
u = 0 x ∈ @
,
where △pu = div(|∇u|p−2∇u) is the p-Laplacian of u,
 is a bounded domain in Rn(n ≥ 3),
1 < p < n, p = pn
n−p is the critical exponent for the Sobolev imbedding,  > 0 and f(x, u)
satisfies some conditions. It reaches the conclusion that this problem has infinitely many
solutions. Some results as p = 2 or f(x, u) = |u|q−2u, where 1 < q < p, are generalized.

Key words: critical Sobolev exponent;concentration compactness principle;genus, infi-
nitely many solutions

中图分类号: 

  • 35J65