Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions

Miyamoto, Yasuhito

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 1, pp. 59-81.

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

$${ϵ}^2\Delta u-u+u^p=0\text{in}A,{\partial }_{\nu }u=0\text{on}\partial A.$$

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

Classification : 35J91, 35B32, 35B25, 35P15
Mots clés : Symmetry breaking bifurcation, Asymptotic transversality, Singular perturbation, Boundary concentration, Nonradially symmetric solutions

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