Singularly perturbed elliptic problems in the case of exchange of stabilities

We consider the singularly perturbed boundary value problem (Eg/epsilon (2) Deltau = f(u,x,epsilon) for x is an element of D. partial derivativeu/partial derivativen - lambda (x) u = 0 for x is an element of Gamma where D subset of R-2 is an open bounded simply connected region with smooth boundary Gamma, epsilon is a small positive parameter and partial derivative'partial derivativen is the derivative along the inner normal of Gamma. We assume that the degenerate problem (E-0) f(u,x,0) = 0 has two solutions p(1)(x) and p(2)(x) intersecting in an smooth Jordan curve C located in D such that f(u)(p(1)(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 epsilon, problem (E-epsilon) has at least one solution u(x,epsilon) satisfying x(x,epsilon) less than or equal to u(x,epsilon) less than or equal to beta (x,epsilon) where the upper and lower solutions beta (x,epsilon) and x(x,epsilon) respectively fulfil beta (x,epsilon) - x(x,epsilon) = O(root epsilon) for x in a delta -neighborhood of C where delta is any fixed positive number sufficiently small, while beta (x,epsilon) - x(x,epsilon) = O(epsilon) for x is an element of (D) over bar \D-delta. In case that f does not depend on epsilon 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. (C) 2001 Academic Press.