Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions

Miyamoto, Yasuhito

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 59-81 / Harvested from Numdam

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Résumé

Let $A : = {a < | x | < 1 + a} \subset ℝ^{N}$ and $p ⩾ 2$ . We consider the Neumann problem $ϵ^{2} Δ u - u + u^{p} = 0 in A, \partial_{ν} u = 0 on \partial A .$ 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 $(0, 1) \times (0, a) \subset ℝ^{2}$ , 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.

Publié le : 2012-01-01

MR 2876247 | Zbl 1241.35104

DOI : https://doi.org/10.1016/j.anihpc.2011.09.003
Classification: 35J91, 35B32, 35B25, 35P15

@article{AIHPC_2012__29_1_59_0,
     author = {Miyamoto, Yasuhito},
     title = {Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {29},
     year = {2012},
     pages = {59-81},
     doi = {10.1016/j.anihpc.2011.09.003},
     mrnumber = {2876247},
     zbl = {1241.35104},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_1_59_0}
}

Miyamoto, Yasuhito. Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 59-81. doi : 10.1016/j.anihpc.2011.09.003. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_1_59_0/

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