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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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[2] H. Berestycki and P.L. Lions , ‘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).

[5] E.N. Dancer , ‘On the number of positive solutions of some weakly nonlinear equations in annular domains’, Math. Z 206 (1991), 551562.

[7] B. Gidas , W.M. Ni and L. Nirenberg , ‘Symmetry and related properties by the maximum principle’, Comm. Math. Phys. 68 (1979), 209243.

[8] B. Gidas and J. Spruck , ‘A priori bounds for positive solutions of nonlinear elliptic equations’, Comm. Partial Differential Equations 6 (1981), 883901.

[9] B. Gidas and J. Spruck , ‘Global and local behaviour of positive solutions of nonlinear elliptic equations’, Comm. Pure Appl. Math. 34 (1981), 525598.

[10] D. Gilbarg and N. Trudinger , Elliptic partial differential equations of second order (Springer-Verlag, Berlin, Heidelberg, New York, 1977).

[11] R. Holmes , Geometric functional analysis and applications (Springer-Verlag, Berlin, Heidelberg, New York, 1975).

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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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