数学物理学报(英文版) ›› 2001, Vol. 21 ›› Issue (2): 203-212.

• 论文 • 上一篇    下一篇

THE EXISTENCE OF MULTIPLE SOLUTIONS OF p-LAPLACIAN ELLIPTIC EQUATION

 谭忠, 姚正安   

  1. Department of Mathematics, Xiamen University, Xiamen 361005, China Department of Mathematics, Zhongshan University, Guangzhou 510275, China
  • 出版日期:2001-04-07 发布日期:2001-04-07
  • 基金资助:

    Supported by CNSF, NSF of Guangdong and Fujian

THE EXISTENCE OF MULTIPLE SOLUTIONS OF p-LAPLACIAN ELLIPTIC EQUATION

 TAN Zhong, YAO Zheng-An   

  1. Department of Mathematics, Xiamen University, Xiamen 361005, China Department of Mathematics, Zhongshan University, Guangzhou 510275, China
  • Online:2001-04-07 Published:2001-04-07
  • Supported by:

    Supported by CNSF, NSF of Guangdong and Fujian

PDF (PC)

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摘要：

In this paper, we consider the quasilinear elliptic equation(− △p u = |u|m−1u + |u|q−1u, x ∈ ,u ∈ W 1,p0 (),(1) Where −△pu = −div(|▽u|p−2▽u), and 0 < m < p−1 < q < +∞,  is a bounded domain in RN(N ≥ 3).  is a positive number. Our object is to estimate exactly the magnitute of
 such that (1) has at least one positive solution if  ∈ (0, ) and no positive solutions if  > . Furthermore, (1) has at least one positive solution when  = , and at least two positive solutions when  ∈ (0, ) and q ≤ Np N−p −1. Finally, we obtain a multiplicity result with positive energy of (1) when 0 < m < p − 1 < q = Np N−p − 1.

关键词: Quasilinear elliptic equation, super- and subsolution method, critical Sobolev exponent, positive solutions, multiple solutions

Abstract:

In this paper, we consider the quasilinear elliptic equation(− △p u = |u|m−1u + |u|q−1u, x ∈ ,u ∈ W 1,p0 (),(1) Where −△pu = −div(|▽u|p−2▽u), and 0 < m < p−1 < q < +∞,  is a bounded domain in RN(N ≥ 3).  is a positive number. Our object is to estimate exactly the magnitute of
 such that (1) has at least one positive solution if  ∈ (0, ) and no positive solutions if  > . Furthermore, (1) has at least one positive solution when  = , and at least two positive solutions when  ∈ (0, ) and q ≤ Np N−p −1. Finally, we obtain a multiplicity result with positive energy of (1) when 0 < m < p − 1 < q = Np N−p − 1.

Key words: Quasilinear elliptic equation, super- and subsolution method, critical Sobolev exponent, positive solutions, multiple solutions

中图分类号: 

  • 35J20
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