On a Sobolev inequality with remainder terms

Authors:
Guozhen Lu and Juncheng Wei

Journal:
Proc. Amer. Math. Soc. **128** (2000), 75-84

MSC (2000):
Primary 35P30, 35J35, 49R50; Secondary 46E35

Published electronically:
September 9, 1999

MathSciNet review:
1694339

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we consider the Sobolev inequality

where is the best Sobolev constant and is the space obtained by taking the completion of with the norm . We prove here a refined version of this inequality,

where is a positive constant, the distance is taken in the Sobolev space , and is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (*A note on the Sobolev inequality*, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (*Sobolev inequalities with remainder terms*, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality

A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation

where and is the unique radial function in with . We will show that the eigenvalues of the above equation are discrete:

and the corresponding eigenfunction spaces are

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Additional Information

**Guozhen Lu**

Affiliation:
Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435

Email:
gzlu@math.wright.edu

**Juncheng Wei**

Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

Email:
wei@math.cuhk.edu.hk

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05497-0

Keywords:
Sobolev inequality,
fourth order equation,
nonlinear eigenvalue problems,
remainder terms

Received by editor(s):
March 3, 1998

Published electronically:
September 9, 1999

Communicated by:
Lesley M. Sibner

Article copyright:
© Copyright 1999
American Mathematical Society