Singular perturbations of an elliptic operator with discontinuous nonlinearity

A family of nonlinear elliptic boundary value problems, affected by the presence of two positive parameters λ and ε, is considered. These problems arise in the theoretical treatment of membranes with enzymic activity, ε being in several realistic situations a small parameter compared to λ. We give an appropriate formulation of the reduced problem which corresponds to ε = 0. On the other hand there exists a critical value λ_C of λ such that for λ ≤ λ_C the problems with ε > 0 are regular and for λ > λ_C singular perturbations of the reduced problem. Some estimates for λ_C as a functional of the boundary function ϕ are derived and an explicit formula for the first variation of λ_C with respect to ϕ is established. Newton-Kantorovich iterative procedure is applied in order to approximate the solutions and to derive some asymptotic formulae. This procedure gives rise to an interesting class of linear coercive singular perturbations with discontinuous coefficients.