Acta mathematica scientia,Series B ›› 2004, Vol. 24 ›› Issue (4): 672-690.

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ON THE SINGULAR VARIATIONAL PROBLEMS

 TAN Jing-Gang, YANG Jian-Fu   

  1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
  • Online:2004-10-20 Published:2004-10-20
  • Supported by:

    Supported by NSFC (10271118) and National Key Program for Basic Research of China(2002CCA03700)

Abstract:

The authors deal with the singular variational problem S(a, b, 0) := inf u∈E,u6≡0 R RN (||x|−a∇u|m + 0|x|−(a+1)m|u|m) dx(R RN ||x|−bu|p dx)m/p
as well as eS = eS(a, b, 1, 2) := infu,v∈E,(u,v)6≡(0,0) R RN J(u, v) dx(R RN |x|−bp|u| |v| dx)m/p where J(u, v) = ||x|−a∇u|m + 1|x|−(a+1)m|u|m + ||x|−a∇v|m + 2|x|−(a+1)m|v|m,N ≥ m + 1 > 2, 0 ≤ a < N−m m , a ≤ b < a + 1 and p = p(a, b) = + =Nm N−m+m(b−a) , , ≥ 1,E = D1,m a (RN). The aim of this paper is to show the existence of minimizer for S(a, b, 0) and eS(a, b, 1, 2).

Key words: Singular variational problems, existemce of minimizer

CLC Number: 

  • 35J35
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