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

Abstract

We consider the singular perturbation problem $-{\epsilon }^2\mathrm{\Delta }u+(u-a(|x|))(u-b(|x|))=0$ in the unit ball of ${\mathbb{R}}^N$, $N⩾1$, 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 $\epsilon \to 0$. 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 $[c+o(1)]{\epsilon }^{-\frac{2}{3}(N-1)}$ as $\epsilon \to 0$, with $c>0$ a certain positive constant. Furthermore, we prove the existence of a decreasing sequence of ${\epsilon }_k>0$, with ${\epsilon }_k\to 0$ as $k\to +\infty$, such that non-radial solutions bifurcate from the unstable branch $\{(u_{-}(\epsilon ),\epsilon ),\epsilon >0\}$ at $\epsilon ={\epsilon }_k$, $k=1,2,\dots$. 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.