Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 1, pp. 59-81.

Let A:={a<|x|<1+a} N and p2. We consider the Neumann problem

ϵ 2 Δu-u+u p =0inA, ν u=0onA.
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) 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.

DOI: 10.1016/j.anihpc.2011.09.003
Classification: 35J91, 35B32, 35B25, 35P15
Keywords: Symmetry breaking bifurcation, Asymptotic transversality, Singular perturbation, Boundary concentration, Nonradially symmetric solutions
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Miyamoto, Yasuhito. Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 1, pp. 59-81. doi : 10.1016/j.anihpc.2011.09.003. http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.003/

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