Singularly perturbed elliptic problems in the case of exchange of stabilities
- Butuzov, Valentin F.
- Nefedov, Nikolai N.
- Schneider, Klaus R.
2010 Mathematics Subject Classification
- 35B25 35J25
- Singular perturbation, asymptotic methods, upper and lower solutions, jumping behavior of reactionrates
We consider the singularly perturbed boundary value problem $(E_ve) , ve^ 2 Delta u = f(u,x,ve)$ for $x in D, , fracpartial upartial n - lambda(x) u =0 quad mboxfor quad x in Gamma$ 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 $beta(x,ve) - alpha(x,ve) = O(sqrtve)$ 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 overlineDbackslash 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.