Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-07-04T17:26:46.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence and boundedness of a minimiser for a constrained minimisation problem on Rn with limiting exponent

Published online by Cambridge University Press:  14 November 2011

Shen Yaotian  and
Yan Shusen
Shen Yaotian
Affiliation:
Department of Applied Mathematics, South China University of Technology, Guangzhou 510641, P.R. China
Yan Shusen
Affiliation:
Department of Applied Mathematics, South China University of Technology, Guangzhou 510641, P.R. China
Get access Rights & Permissions [Opens in a new window]

Synopsis

We use the concentration compactness principle to study the existence of a minimiser of the minimisation problem where u =(u1, …, uN), . We also prove the boundedness of the minimiser of l1 by using the reverse Holder inequality.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 3-4 , 1992 , pp. 221 - 235
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

1Aubin, T.. Problèmes isoperimetriques et éspaces de Sobolev. J. Differential Geom. 11 (1976), 573–598.CrossRefGoogle Scholar
2Brezis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent. Comm. Pure Appl. Math. 36 (1983), 437–477.CrossRefGoogle Scholar
3Ekeland, I.. Nonconvex minimization problem. Bull. Amer. Math. Soc. 1 (1979), 443–474.CrossRefGoogle Scholar
4Giaquinta, M.. Multiple integrals in the calculus of variation and nonlinear elliptic systems (Princeton: Princeton University Press, 1983).Google Scholar
5Giaquinta, M. and Giusti, E.. On the regularity of the minima of variational integrals. Acta Math. 148 (1982), 31–46.CrossRefGoogle Scholar
6Ladyzhenskaya, O. A. and Ural'tseva, N. N.. Linear and quasilinear elliptic equations, 2nd Russian edn. (Moscow: Nauka, 1973).Google Scholar
7Lions, P. L.. The concentration compactness principle in the calculus of variations. The locally compact case. Ann. Inst. H. Poincaré Anal. Non-Linéaire, Part I, 1 (1984), 109–145; Part II, 1 (1984), 223–283.CrossRefGoogle Scholar
8Lions, P. L.. The concentration compactness principle in the calculus of variations. The limit case, Part I. Rev. Mat. Iberoamericana 1 (1985), 145–201.CrossRefGoogle Scholar
9Yaotian, Shen and Shusen, Yan. Eigenvalue problem for quasilinear elliptic systems with limiting nonlinearity (to appear).Google Scholar
10Struwe, M.. Quasilinear elliptic eigenvalue problems. Comment. Math. Helv. 58 (1983), 509–527.CrossRefGoogle Scholar
11Talenti, G.. Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372.CrossRefGoogle Scholar
12Shusen, Yan and Gongbao, Li. A minimization problem involving critical Sobolev exponent and its related Euler-Lagrange equation. Arch. Rational Mech. Anal. 114 (1991), 365–381.CrossRefGoogle Scholar