# Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities

### Georgia Karali

Department of Applied Mathematics, University of Crete, GR-714 09 Heraklion, Crete, Greece, Institute of Applied and Computational Mathematics, IACM, FORTH, Greece### Christos Sourdis

Department of Applied Mathematics, University of Crete, GR-714 09 Heraklion, Crete, Greece

## Abstract

We consider the singular perturbation problem $−ε_{2}Δu+(u−a(∣x∣))(u−b(∣x∣))=0$ in the unit ball of $R_{N}$, $N⩾1$, 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 $ε→0$. 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 $[c+o(1)]ε_{−32(N−1)}$ as $ε→0$, 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.

## Cite this article

Georgia Karali, Christos Sourdis, Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 2, pp. 131–170

DOI 10.1016/J.ANIHPC.2011.09.005