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

Full-text PDF Free Access

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.**Gabriele Bianchi and Henrik Egnell,*A note on the Sobolev inequality*, J. Funct. Anal.**100**(1991), no. 1, 18–24. MR**1124290**, 10.1016/0022-1236(91)90099-Q**2.**Haïm Brezis and Elliott H. Lieb,*Sobolev inequalities with remainder terms*, J. Funct. Anal.**62**(1985), no. 1, 73–86. MR**790771**, 10.1016/0022-1236(85)90020-5**3.**Reinhold G. Meise and Dietmar Vogt,*An interpretation of 𝜏_{𝜔} and 𝜏_{𝛿} as normal topologies of sequence spaces*, Functional analysis, holomorphy and approximation theory (Rio de Janeiro, 1980) North-Holland Math. Stud., vol. 71, North-Holland, Amsterdam-New York, 1982, pp. 273–285. MR**691168****4.**D. E. Edmunds, D. Fortunato, and E. Jannelli,*Critical exponents, critical dimensions and the biharmonic operator*, Arch. Rational Mech. Anal.**112**(1990), no. 3, 269–289. MR**1076074**, 10.1007/BF00381236**5.**Henrik Egnell, Filomena Pacella, and Mariarosaria Tricarico,*Some remarks on Sobolev inequalities*, Nonlinear Anal.**13**(1989), no. 6, 671–681. MR**998512**, 10.1016/0362-546X(89)90086-2**6.**P.-L. Lions,*The concentration-compactness principle in the calculus of variations. The limit case. I*, Rev. Mat. Iberoamericana**1**(1985), no. 1, 145–201. MR**834360**, 10.4171/RMI/6

P.-L. Lions,*The concentration-compactness principle in the calculus of variations. The limit case. II*, Rev. Mat. Iberoamericana**1**(1985), no. 2, 45–121. MR**850686**, 10.4171/RMI/12**7.**Chang-Shou Lin,*A classification of solutions of a conformally invariant fourth order equation in 𝑅ⁿ*, Comment. Math. Helv.**73**(1998), no. 2, 206–231. MR**1611691**, 10.1007/s000140050052**8.**Giorgio Talenti,*Best constant in Sobolev inequality*, Ann. Mat. Pura Appl. (4)**110**(1976), 353–372. MR**0463908****9.**Xu Jia Wang,*Sharp constant in a Sobolev inequality*, Nonlinear Anal.**20**(1993), no. 3, 261–268. MR**1202203**, 10.1016/0362-546X(93)90162-L

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