Resumen
We consider the singularly perturbed boundary value problem (E) 2 u=f (u, x, ) for xD, - (x) u=0 for x where DR2 is an open bounded simply connected region with smooth boundary , is a small positive parameter and /n is the derivative along the inner normal of . 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 (x, )u (x, ) (x, ) where the upper and lower solutions (x, ) and (x, ) respectively fulfil (x, )- (x, )=O () for x in a -neighborhood of where is any fixed positive number sufficiently small, while (x, )- (x, )=O () for xD\D. 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.