Singularly Perturbed Elliptic Problems in the Case of Exchange of Stabilities

We consider the singularly perturbed boundary value problem (E ) 2 Δu=f(u, x, ) for x D, −λ(x) u=0 for x Γ where D R2 is an open bounded simply connected region with smooth boundary Γ, is a small positive parameter and ∂/∂n is the derivative along the inner normal of Γ. We assume that the degenerate problem (E0) f(u, x, 0)=0 has two solutions 1(x) and 2(x) intersecting in an smooth Jordan curve located in D such that fu( i(x), x, 0) changes its sign on for i=1, 2 (exchange of stabilities). By means of the method of asymptotic lower and upper solutions we prove that for sufficiently small , problem (E ) has at least one solution u(x, ) satisfying α(x, ) u(x, ) β(x, ) where the upper and lower solutions β(x, ) and α(x, ) respectively fulfil β(x, )−α(x, )=O( ) for x in a δ-neighborhood of where δ is any fixed positive number sufficiently small, while β(x, )−α(x, )=O( ) for x D\Dδ. In case that f does not depend on these estimates can be improved. Applying this result to a special reaction system in a nonhomogeneous medium we prove that the reaction rate exhibits a spatial jumping behavior