Singularly Perturbed Elliptic Problems in the Case of Exchange of Stabilities

Abstract

We consider the singularly perturbed boundary value problem (Eε)ε2Δu=f(u, x, ε) for x∈D, ∂u∂n−λ(x)u=0 for x∈Γ where D⊂R2 is an open bounded simply connected region with smooth boundary Γ, ε 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 ϕ1(x) and ϕ2(x) intersecting in an smooth Jordan curve C located in D such that fu(ϕi(x), x, 0) changes its sign on 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 ε, problem (Eε) has at least one solution u(x, ε) satisfying α(x, ε)⩽u(x, ε)⩽β(x, ε) where the upper and lower solutions β(x, ε) and α(x, ε) respectively fulfil β(x, ε)−α(x, ε)=O(ε) for x in a δ-neighborhood of C where δ is any fixed positive number sufficiently small, while β(x, ε)−α(x, ε)=O(ε) for x∈D\\Dδ. In case that f does not depend on ε 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.