The steepest point of the boundary layers of singularly perturbed semilinear elliptic problems

We consider the nonlinear singularly perturbed problem -epsilon(2)Deltau = f(u), u > 0 in Omega, u = 0 on partial derivativeOmega, where Omega subset of R-N (N greater than or equal to 2) is an appropriately smooth bounded domain and epsilon > 0 is a small parameter. It is known that under some conditions on f, the solution u, corresponding to epsilon develops boundary layers when epsilon --> 0. We determine the steepest point of the boundary layers on the boundary by establishing an asymptotic formula for the slope of the boundary layers with exact second term.