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

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

Published electronically:
September 9, 1999

MathSciNet review:
1694339

Full-text PDF

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

**1.**G. Bianchi and H. Egnell, A note on the Sobolev inequality,*J. Funct. Anal.*100 (1991), 18-24. MR**92i:46033****2.**H. Brezis and E. Lieb, Sobolev inequalities with remainder terms,*J. Funct. Anal.*62 (1985), 73-86. MR**86i:46033****3.**H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,*Comm. Pure Appl. Math.*36 (1983), 437-477. MR**84h:46053****4.**D.E. Edmunds, D. Fortunato and E. Janelli, Critical exponents, critical dimensions, and the biharmonic operator,*Arch. Rational Mech. Anal.*112 (1990), 269-289. MR**91k:35191****5.**H. Egnell, F. Pacella and M. Tricarico, Some remarks on Sobolev inequalities,*Nonlinear Anal. T.M.A.*13 (1989), 671-681. MR**90h:46061****6.**P. L. Lions, The concentration-compactness principle in the calculus of variations: the limiting case (parts 1 and 2),*Riv. Mat. Iberoamericana*1 (1985), 145-201, 45-121. MR**87c:49007**; MR**87j:49012****7.**C.S. Lin, A classification of solutions of a conformally invariant fourth order equation in , Comment. Math. Helv.**73**(1998), 206-231. MR**99c:35062****8.**G. Talenti, Best constant in Sobolev inequalities,*Ann. Mat. Pura Appl.*110 (1976), 353-372. MR**57:3846****9.**X. Wang, Sharp constant in a Sobolev inequality,*Nonlinear Analysis: TMA*, 20 (1993), 261-268. MR**94g:35035**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35P30,
35J35,
49R50,
46E35

Retrieve articles in all journals with MSC (2000): 35P30, 35J35, 49R50, 46E35

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:
https://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