Hostname: page-component-7dd5485656-bvgqh Total loading time: 0 Render date: 2025-10-31T13:58:21.821Z Has data issue: false hasContentIssue false
  • English
  • Français

Some notes on the method of moving planes

Published online by Cambridge University Press:  17 April 2009

E.N. Dancer
E.N. Dancer
Affiliation:
The Department of Mathematics, Statistics and Computing Science, The University of New England, Armidale NSW 2351, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 3 , December 1992 , pp. 425 - 434
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Berestycki, H. and Nirenberg, L., ‘On the method of moving planes and the sliding method’, (preprint), École. Normals Superieure (1991).CrossRefGoogle Scholar
[2]Berestycki, H. and Lions, P.L., ‘Some applications of the method of sub and super solutions’, in Bifurcation and nonlinear eigenvalue problems, Lecture Notes in Math. 782, pp. 16–41 (Springer-Verlag, Berlin, Heidelberg, New York, 1980).CrossRefGoogle Scholar
[3]Church, P., Dancer, E.N. and Timourian, J., ‘The structure of a nonlinear elliptic operator’, Trans. Amer. Math. Soc. (to appear).Google Scholar
[4]Dancer, E.N., ‘Weakly nonlinear Dirichlet problems on long or thin domains’, Mem. Amer. Math. Soc. (1991) (to appear).Google Scholar
[5]Dancer, E.N., ‘On the number of positive solutions of some weakly nonlinear equations in annular domains’, Math. Z 206 (1991), 551–562.CrossRefGoogle Scholar
[6]Dancer, E.N., ‘On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large’, Proc. London Mat. Soc. 53 (1986), 429–452.CrossRefGoogle Scholar
[7]Gidas, B., Ni, W.M. and Nirenberg, L., ‘Symmetry and related properties by the maximum principle’, Comm. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
[8]Gidas, B. and Spruck, J., ‘A priori bounds for positive solutions of nonlinear elliptic equations’, Comm. Partial Differential Equations 6 (1981), 883–901.CrossRefGoogle Scholar
[9]Gidas, B. and Spruck, J., ‘Global and local behaviour of positive solutions of nonlinear elliptic equations’, Comm. Pure Appl. Math. 34 (1981), 525–598.CrossRefGoogle Scholar
[10]Gilbarg, D. and Trudinger, N., Elliptic partial differential equations of second order (Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[11]Holmes, R., Geometric functional analysis and applications (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[12]Kinderlehrer, D. and Stampacchia, G., An introduction to variational inequalities and their applications (Academic Press, New York, 1980).Google Scholar
[13]Protter, M. and Weinberger, H., Maximum principles in differential equations (Prentice Hall, Englewood Cliffs, 1967).Google Scholar