We consider the singular perturbation problem in the unit ball of , , under Neumann boundary conditions. The assumption that changes sign in , known as the case of exchange of stabilities, is the main source of difficulty. More precisely, under the assumption that has one simple zero in , we prove the existence of two radial solutions and that converge uniformly to , as . The solution is asymptotically stable, whereas has Morse index one, in the radial class. If , we prove that the Morse index of , in the general class, is asymptotically given by as , with a certain positive constant. Furthermore, we prove the existence of a decreasing sequence of , with as , such that non-radial solutions bifurcate from the unstable branch at , . 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.
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}, pages = {131--170}, publisher = {Elsevier}, volume = {29}, number = {2}, year = {2012}, doi = {10.1016/j.anihpc.2011.09.005}, zbl = {1242.35114}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.005/} }
TY - JOUR AU - Karali, Georgia AU - Sourdis, Christos TI - Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 SP - 131 EP - 170 VL - 29 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.005/ DO - 10.1016/j.anihpc.2011.09.005 LA - en ID - AIHPC_2012__29_2_131_0 ER -
%0 Journal Article %A Karali, Georgia %A Sourdis, Christos %T Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 131-170 %V 29 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.005/ %R 10.1016/j.anihpc.2011.09.005 %G en %F 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/articles/10.1016/j.anihpc.2011.09.005/
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