Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions

Abstract

Let $A:=\{a<|x|<1+a\}\subset {\mathbb{R}}^N$ and $p⩾2$. We consider the Neumann problem

Let $\lambda =1/{\epsilon }^2$. When $\lambda$ is large, we prove the existence of a smooth curve $\{(\lambda ,u(\lambda ))\}$ 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 $(0,1)\times (0,a)\subset {\mathbb{R}}^2$, we show that a curve consisting of one-dimensional solutions concentrating on $\{0\}\times [0,a]$ has infinitely many symmetry breaking bifurcation points. Extending this solution with even reflection, we obtain a new entire solution.