Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T02:07:32.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large

Published online by Cambridge University Press:  14 November 2011

Zongming Guo  and
J. R. L. Webb
Zongming Guo
Affiliation:
Department of Mathematics, University of New England, Armidale, NSW23 51, Australia
J. R. L. Webb
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

Existence and uniqueness results are proved for positive solutions of a class of quasilinear elliptic equations in a domain Ω⊂ℝN via a generalisation of Serrin's sweeping principle. In the case when Ω is an annulus, it is shown that the solution is radially symmetric.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 1 , 1994 , pp. 189 - 198
Copyright
Copyright © Royal Society of Edinburgh 1994

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

1Angenent, S. B.. Uniqueness of the solution of a semilinear boundary value problem. Math. Ann. 272 (1985), 129–138.CrossRefGoogle Scholar
2Boccardo, L., Drabek, P. and Kucera, M.. Landesman-Laser conditions for strongly nonlinear boundary value problems. Comment. Math. Univ. Carolinae 30 (1989), 411–427.Google Scholar
3Dancer, E. N.. Uniqueness for elliptic equations when a parameter is large. Nonlinear Anal. 8 (1984), 835–836.CrossRefGoogle Scholar
4Dancer, E. N.. On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large. Proc. London Math. Soc. (3) 53 (1986), 429–452.CrossRefGoogle Scholar
5Gidas, B., Ni, W. M. and Nirenberg, L.. Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
6Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (Berlin: Springer, 1977).CrossRefGoogle Scholar
7Gossez, J. P.. Strongly nonlinear elliptic problems near resonance: A variational approach. Comm. Partial Differential Equations 15 (1990), 1141–1159.Google Scholar
8Guedda, M. and Veron, L.. Local and global properties of solutions of quasilinear elliptic equations. J. Differential Equations 76 (1988), 159–189.CrossRefGoogle Scholar
9Guo, Z. M.. Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems. Nonlinear Anal. 18 (1992), 957–971.CrossRefGoogle Scholar
10Guo, Z. M.. Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations. Applicable Anal. 47 (1992), 173–190.CrossRefGoogle Scholar
11Kichenassamy, S. and Smoller, J.. On the existence of radial solutions of quasilinear elliptic equations. Nonlinearity 3 (1990), 677–694.CrossRefGoogle Scholar
12Lin, S. S.. On the number of positive solutions for nonlinear elliptic equations when a parameter is large. Nonlinear Anal. 16 (1991), 283–297.CrossRefGoogle Scholar
13Pino, M. D., Elgueta, M. and Mamasevich, R.. A homotopic deformation along p of a Leray-Schauder degree result and existence for (|u″p−2u″) + f(t, u) = 0, u(0) = u(T) = 0,p>l. J. Differential Equations 80(1989), 1–13.CrossRefGoogle Scholar
14Sakaguchi, S.. Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 14 (1987), 403–421.Google Scholar