Boundary and interior layer behavior of solutions of some singularly perturbed semilinear elliptic boundary value problems (Q800563)

We study the asymptotic behavior (as \(\epsilon\to 0)\) of solutions of the singularly perturbed semilinear elliptic boundary value problem \((*)\quad\epsilon^ 2\Delta u=h(\underset \tilde{} x,u)\) in D, \(u(\underset \tilde{} x,\epsilon)=f(\underset \tilde{} x)\) on B, where \(\Delta\) is the Laplacian operator, D is an open set in \(R^ n\) bounded by the smooth manifold B, and h and f are prescribed smooth functions. Using a method based on the theory of partial differential inequalities we show that under explicit conditions on h and f the problem (*) has a solution which exhibits boundary layer behavior and one of three distinct types of interior layer behavior. The conditions on h and f are given in terms of a potential energy functional \(V(\underset \tilde{} x,u)=\int^{u}_{0}h(\underset \tilde{} x,s)ds.\) Our method of proof also allows the construction of approximate solutions of problem (*) which serve as lower and upper bounds for the exact solutions.