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Some notes on the method of moving planes

  • E.N. Dancer (a1)
Abstract

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.

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References
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[1]Berestycki, H. and Nirenberg, L., ‘On the method of moving planes and the sliding method’, (preprint), École. Normals Superieure (1991).
[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. 1641 (Springer-Verlag, Berlin, Heidelberg, New York, 1980).
[3]Church, P., Dancer, E.N. and Timourian, J., ‘The structure of a nonlinear elliptic operator’, Trans. Amer. Math. Soc. (to appear).
[4]Dancer, E.N., ‘Weakly nonlinear Dirichlet problems on long or thin domains’, Mem. Amer. Math. Soc. (1991) (to appear).
[5]Dancer, E.N., ‘On the number of positive solutions of some weakly nonlinear equations in annular domains’, Math. Z 206 (1991), 551562.
[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), 429452.
[7]Gidas, B., Ni, W.M. and Nirenberg, L., ‘Symmetry and related properties by the maximum principle’, Comm. Math. Phys. 68 (1979), 209243.
[8]Gidas, B. and Spruck, J., ‘A priori bounds for positive solutions of nonlinear elliptic equations’, Comm. Partial Differential Equations 6 (1981), 883901.
[9]Gidas, B. and Spruck, J., ‘Global and local behaviour of positive solutions of nonlinear elliptic equations’, Comm. Pure Appl. Math. 34 (1981), 525598.
[10]Gilbarg, D. and Trudinger, N., Elliptic partial differential equations of second order (Springer-Verlag, Berlin, Heidelberg, New York, 1977).
[11]Holmes, R., Geometric functional analysis and applications (Springer-Verlag, Berlin, Heidelberg, New York, 1975).
[12]Kinderlehrer, D. and Stampacchia, G., An introduction to variational inequalities and their applications (Academic Press, New York, 1980).
[13]Protter, M. and Weinberger, H., Maximum principles in differential equations (Prentice Hall, Englewood Cliffs, 1967).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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