Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions
Yasuhito Miyamoto · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2011
Let A: = \{a < |x| < 1 + a\} \subset \mathbb{R}^{N} and p⩾2 . We consider the Neumann problem \varepsilon ^{2}\mathrm{\Delta }u−u + u^{p} = 0\:\text{in }A,\:\partial _{ u }u = 0\:\text{on }\partial A. Let \lambda = 1/ \varepsilon ^{2} . When λ 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.