# 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 $A:={a<∣x∣<1+a}⊂R_{N}$ and $p⩾2$. We consider the Neumann problem

Let $λ=1/ε_{2}$. When *λ* is large, we prove the existence of a smooth curve ${(λ,u(λ))}$ consisting of radially symmetric and radially decreasing solutions concentrating on ${∣x∣=a}$. 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 $N=2$, then the closure of each bifurcating continuum is locally homeomorphic to a disk. When the domain is a rectangle $(0,1)×(0,a)⊂R_{2}$, we show that a curve consisting of one-dimensional solutions concentrating on ${0}×[0,a]$ 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