A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain

The semilinear reaction-di®usion equation ¡"24u+b(x; u) = 0 with Dirichlet bound-ary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the “reduced equation” b(x, u0 (x)) = 0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer functions. By perturbing this
asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation −Dz + f (z) = 0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.