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

Karali, Georgia; Sourdis, Christos

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 131-170.

We consider the singular perturbation problem $-{ϵ}^2\Delta u+(u-a(\left|x\right|\left)\right)(u-b(\left|x\right|\left)\right)=0$ 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 $(0,1)$, 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 $ϵ\to 0$. 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 $[c+o\left(1\right)]{ϵ}^{-\frac{2}{3}(N-1)}$ 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 $\{(u_{-}\left(ϵ\right),ϵ),ϵ>0\}$ 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.

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