Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, p. 131-170
We consider the singular perturbation problem -ϵ 2 Δu+(u-a(|x|))(u-b(|x|))=0 in the unit ball of N , N1, under Neumann boundary conditions. The assumption that a(r)-b(r) changes sign in (0,1), known as the case of exchange of stabilities, is the main source of difficulty. More precisely, under the assumption that a-b has one simple zero in (0,1), we prove the existence of two radial solutions u + and u - that converge uniformly to max {a,b}, as ϵ0. The solution u + is asymptotically stable, whereas u - has Morse index one, in the radial class. If N2, we prove that the Morse index of u - , in the general class, is asymptotically given by [c+o(1)]ϵ -2 3(N-1) as ϵ0, with c>0 a certain positive constant. Furthermore, we prove the existence of a decreasing sequence of ϵ k >0, with ϵ k 0 as k+, such that non-radial solutions bifurcate from the unstable branch {(u - (ϵ),ϵ),ϵ>0} at ϵ=ϵ k , k=1,2,. Our approach is perturbative, based on the existence and non-degeneracy of solutions of a “limit” problem. Moreover, our method of proof can be generalized to treat, in a unified manner, problems of the same nature where the singular limit is continuous but non-smooth.
DOI : https://doi.org/10.1016/j.anihpc.2011.09.005
Classification:  35J25,  35J20,  35B33,  35B40
Keywords: Corner layer, Exchange of stabilities, Geometric singular perturbation theory, Non-radial bifurcations
@article{AIHPC_2012__29_2_131_0,
     author = {Karali, Georgia and Sourdis, Christos},
     title = {Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {29},
     number = {2},
     year = {2012},
     pages = {131-170},
     doi = {10.1016/j.anihpc.2011.09.005},
     zbl = {1242.35114},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2012__29_2_131_0}
}
Karali, Georgia; Sourdis, Christos. Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, pp. 131-170. doi : 10.1016/j.anihpc.2011.09.005. http://www.numdam.org/item/AIHPC_2012__29_2_131_0/

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