A global compactness result for singular elliptic problems involving critical Sobolev exponent

Authors:
Daomin Cao and Shuangjie Peng

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1857-1866

MSC (2000):
Primary 35J60; Secondary 35B33

Published electronically:
October 1, 2002

MathSciNet review:
1955274

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a bounded domain such that . Let be a (P.S.) sequence of the functional . We study the limit behaviour of and obtain a global compactness result.

**[1]**Adimurthi and Michael Struwe,*Global compactness properties of semilinear elliptic equations with critical exponential growth*, J. Funct. Anal.**175**(2000), no. 1, 125–167. MR**1774854**, 10.1006/jfan.2000.3602**[2]**Haïm Brézis and Elliott Lieb,*A relation between pointwise convergence of functions and convergence of functionals*, Proc. Amer. Math. Soc.**88**(1983), no. 3, 486–490. MR**699419**, 10.1090/S0002-9939-1983-0699419-3**[3]**Haïm Brézis and Louis Nirenberg,*Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents*, Comm. Pure Appl. Math.**36**(1983), no. 4, 437–477. MR**709644**, 10.1002/cpa.3160360405**[4]**F. Catrina and Z.Q. Wang,*On the Caffarelli-Kohn-Nirenberg inequalities:sharp constants, existence(and nonexistence), and symmetry of extremal functions*, Comm. Pure Appl. Math.,**53**(2000), 1-30.**[5]**Kai Seng Chou and Chiu Wing Chu,*On the best constant for a weighted Sobolev-Hardy inequality*, J. London Math. Soc. (2)**48**(1993), no. 1, 137–151. MR**1223899**, 10.1112/jlms/s2-48.1.137**[6]**Jean-Michel Coron,*Topologie et cas limite des injections de Sobolev*, C. R. Acad. Sci. Paris Sér. I Math.**299**(1984), no. 7, 209–212 (French, with English summary). MR**762722****[7]**Wei Yue Ding,*Positive solutions of Δ𝑢+𝑢^{(𝑛+2)/(𝑛-2)}=0 on contractible domains*, J. Partial Differential Equations**2**(1989), no. 4, 83–88. MR**1027983****[8]**Henrik Egnell,*Elliptic boundary value problems with singular coefficients and critical nonlinearities*, Indiana Univ. Math. J.**38**(1989), no. 2, 235–251. MR**997382**, 10.1512/iumj.1989.38.38012**[9]**J. P. García Azorero and I. Peral Alonso,*Hardy inequalities and some critical elliptic and parabolic problems*, J. Differential Equations**144**(1998), no. 2, 441–476. MR**1616905**, 10.1006/jdeq.1997.3375**[10]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****[11]**Enrico Jannelli,*The role played by space dimension in elliptic critical problems*, J. Differential Equations**156**(1999), no. 2, 407–426. MR**1705383**, 10.1006/jdeq.1998.3589**[12]**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**[13]**Dario Pierotti and Susanna Terracini,*On a Neumann problem with critical exponent and critical nonlinearity on the boundary*, Comm. Partial Differential Equations**20**(1995), no. 7-8, 1155–1187. MR**1335747**, 10.1080/03605309508821128**[14]**Michael Struwe,*A global compactness result for elliptic boundary value problems involving limiting nonlinearities*, Math. Z.**187**(1984), no. 4, 511–517. MR**760051**, 10.1007/BF01174186**[15]**Susanna Terracini,*On positive entire solutions to a class of equations with a singular coefficient and critical exponent*, Adv. Differential Equations**1**(1996), no. 2, 241–264. MR**1364003****[16]**S.Yan,*A global compactness result for quasilinear elliptic boundary value problems involving limiting nonlinearities*, Chinese Ann. Math.,**16A**(1995), 397-402.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35J60,
35B33

Retrieve articles in all journals with MSC (2000): 35J60, 35B33

Additional Information

**Daomin Cao**

Affiliation:
Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China

Email:
dmcao@mail.amt.ac.cn

**Shuangjie Peng**

Affiliation:
Department of Mathematics, Xiao Gan University, Xiao Gan, People’s Republic of China – and – Institute of Applied Mathematics, AMSS., Chinese Academy of Sciences, Beijing 100080, People’s Republic of China

Email:
pengsj@mail.amss.ac.cn

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06729-1

Keywords:
Palais-Smale sequence,
compactness,
Sobolev and Hardy critical exponents

Received by editor(s):
December 2, 2001

Received by editor(s) in revised form:
January 31, 2002

Published electronically:
October 1, 2002

Additional Notes:
The first author was supported by Special Funds For Major States Basic Research Projects of China (G1999075107) and Knowledge Innovation Funds of CAS in China.

The second author was supported by Knowledge Innovation Funds of CAS in China

Communicated by:
David S. Tartakoff

Article copyright:
© Copyright 2002
American Mathematical Society