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, Tome 29 (2012) no. 2, pp. 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 : 10.1016/j.anihpc.2011.09.005
Classification : 35J25, 35J20, 35B33, 35B40
Mots clés : Corner layer, Exchange of stabilities, Geometric singular perturbation theory, Non-radial bifurcations
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     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},
     pages = {131--170},
     publisher = {Elsevier},
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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, Tome 29 (2012) no. 2, pp. 131-170. doi : 10.1016/j.anihpc.2011.09.005. http://archive.numdam.org/articles/10.1016/j.anihpc.2011.09.005/

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