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 $-{ϵ}^{2}\Delta u+\left(u-a\left(|x|\right)\right)\left(u-b\left(|x|\right)\right)=0$ in the unit ball of ${ℝ}^{N}$, $N⩾1$, under Neumann boundary conditions. The assumption that $a\left(r\right)-b\left(r\right)$ changes sign in $\left(0,1\right)$, 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 $\left(0,1\right)$, we prove the existence of two radial solutions ${u}_{+}$ and ${u}_{-}$ that converge uniformly to $\mathrm{max}\left\{a,b\right\}$, as $ϵ\to 0$. The solution ${u}_{+}$ is asymptotically stable, whereas ${u}_{-}$ has Morse index one, in the radial class. If $N⩾2$, we prove that the Morse index of ${u}_{-}$, in the general class, is asymptotically given by $\left[c+o\left(1\right)\right]{ϵ}^{-\frac{2}{3}\left(N-1\right)}$ as $ϵ\to 0$, with $c>0$ a certain positive constant. Furthermore, we prove the existence of a decreasing sequence of ${ϵ}_{k}>0$, with ${ϵ}_{k}\to 0$ as $k\to +\infty$, such that non-radial solutions bifurcate from the unstable branch $\left\{\left({u}_{-}\left(ϵ\right),ϵ\right),\phantom{\rule{0.166667em}{0ex}}ϵ>0\right\}$ at $ϵ={ϵ}_{k}$, $k=1,2,\cdots$. 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
Mots clés : 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://archive.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, Tome 29 (2012) no. 2, pp. 131-170. doi : 10.1016/j.anihpc.2011.09.005. http://archive.numdam.org/item/AIHPC_2012__29_2_131_0/

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