Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions

  • Yasuhito Miyamoto

    Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo, 152-8551, Japan

Abstract

Let and . We consider the Neumann problem

Let . When λ is large, we prove the existence of a smooth curve consisting of radially symmetric and radially decreasing solutions concentrating on . Moreover, checking the transversality condition, we show that this curve has infinitely many symmetry breaking bifurcation points from which continua consisting of nonradially symmetric solutions emanate. If , then the closure of each bifurcating continuum is locally homeomorphic to a disk. When the domain is a rectangle , we show that a curve consisting of one-dimensional solutions concentrating on has infinitely many symmetry breaking bifurcation points. Extending this solution with even reflection, we obtain a new entire solution.

Cite this article

Yasuhito Miyamoto, Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 1, pp. 59–81

DOI 10.1016/J.ANIHPC.2011.09.003