Singularly Perturbed Elliptic Problems in the Case of Exchange of Stabilities

We consider the singularly perturbed boundary value problem $(E_\ve) \, \ve^ 2 \Delta u = f(u,x,\ve)$ {for} $x \in D, \, \frac{\partial u}{\partial n} - \lambda(x) u =0 \quad \mbox{for} \quad x \in \Gamma$ where $ D \subset R^ 2$ is an open bounded simply connected region with smooth boundary $\Gamma$, $\ve$ is a small positive parameter and $\partial/\partial n$ is the derivative along the inner normal of $\Gamma$. We assume that the degenerate problem $(E_0) \quad f(u,x,0) =0$ has two solutions $\varphi_1(x)$ and $\varphi_2(x)$ intersecting in an smooth Jordan curve ${ \cal C}$ located in $D$ such that $f_u(\varphi_i(x),x,0)$ changes its sign on ${\cal C}$ for $i=1,2$ (exchange of stabilities). By means of the method of asymptotic lower and upper solutions we prove that for sufficiently small $\ve$, problem $(E_\ve)$ has at least one solution $u(x,\ve)$ satisfying $\alpha(x,\ve) \le u(x,\ve) \le \beta(x,\ve) $ where the upper and lower solutions $\beta(x,\ve)$ and $\alpha(x,\ve)$ respectively fulfil $\beta(x,\ve) - \alpha(x,\ve) = O(\sqrt{\ve})$ for $x$ in a $\delta$-neighborhood of ${\cal C}$ where $\delta$ is any fixed positive number sufficiently small, while $\beta(x,\ve) - \alpha(x,\ve) = O(\ve)$ for $ x \in \overline{D}\backslash D_\delta$. Applying this result to a special reaction system in a nonhomogeneous medium we prove that the reaction rate exhibits a spatial jumping behavior