Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-27T21:32:36.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical semilinear biharmonic equations in RN

Published online by Cambridge University Press:  14 November 2011

Ezzat S. Noussair ,
Charles A. Swanson  and
Yang Jianfu
Ezzat S. Noussair
Affiliation:
School of Mathematics, University of New South Wales, Kensington, N.S.W. 2033, Australia
Charles A. Swanson
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Y4
Yang Jianfu
Affiliation:
Department of Mathematics, Jiangxi University, Nanchang, Jiangxi 330047, People's Republic of China
Get access Rights & Permissions [Opens in a new window]

Synopsis

An existence theorem is obtained for a fourth-order semilinear elliptic problem in RN involving the critical Sobolev exponent (N + 4)/(N − 4), N>4. A preliminary result is that the best constant in the Sobolev embedding L2N/(N–4) (RN) is attained by all translations and dilations of (1 + ∣x∣2)(4-N)/2. The best constant is found to be

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 1-2 , 1992 , pp. 139 - 148
Copyright
Copyright © Royal Society of Edinburgh 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory and applications. J. Fund. Anal. 14 (1973), 349–381.CrossRefGoogle Scholar
2Brezis, H. and Lieb, E. H.. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983), 486–490.CrossRefGoogle Scholar
3Brezis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437–477.CrossRefGoogle Scholar
4Edmunds, D. E., Fortunato, D. and Jannelli, E.. Critical exponents, critical dimensions and the biharmonic operator. Arch. Rational Mech. Anal. 112 (1990), 269–289.CrossRefGoogle Scholar
5Egnell, Henrik. Semilinear elliptic equations involving critical Sobolev exponents. Arch. Rational Mech. Anal. 104 (1988), 27–56.CrossRefGoogle Scholar
6Egnell, Henrik. Existence results for some quasilinear elliptic equations. In Proceedings of the conference ‘Variational Problems’, Paris, June 1988, to appear.Google Scholar
7Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order, 2nd edn (Berlin: Springer, 1983).Google Scholar
8Lions, P. L.. The concentration – compactness principle in the Calculus of Variations. The locally compact case, parts 1 & 2. Ann. Inst. H. Poincaré: Anal, non linéaire 1 (1984), 109–145; 223–283.CrossRefGoogle Scholar
9Lions, P. L.. The concentration – compactness principle in the Calculus of Variations. The limit case, Parts 1 & 2. Rev. Mat. Iberoamericana 1(1) (1985), 145–201; 1(2) (1985), 46–120.Google Scholar
10Noussair, E. S. and Swanson, C. A.. Ground states for critical semilinear scalar field equations. Differential and Integral Equations 3 (1990), 875–887.CrossRefGoogle Scholar
11Noussair, E. S., Swanson, C. A. and Jianfu, Yang. Positive finite energy solutions of critical semilinear elliptic problems. Cana d. J. Math., (to appear).Google Scholar
12Rosen, G.. Minimum value for c in the Sobolev inequality ‖ø‖6≦‖∇ø‖2. SIAM J. Appl. Math. 21 (1971), 30–32.CrossRefGoogle Scholar
13Talenti, G.. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372.CrossRefGoogle Scholar
14Dong, Zhang. Positive solutions of nonlinear scalar field equations involving critical Sobolev exponent. Acta Math. Sinica N.S. 3 (1987), 27–37.CrossRefGoogle Scholar