On a class of elliptic boundary-value problems with parameter and discontinuous non-linearity

摘要

We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) for negative (resp, non-negative) values of the phase variable. Let (u) over tilde (x) be a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting v(x) = u(x) - (u) over tilde (x), we reduce the original problem to a problem with homogeneous boundary condition. The spectrum of the transformed problem consists of the values of the parameter for which this problem has a non-zero solution (the function v(x) = 0 is a solution for all values of the parameter). Under certain additional restrictions we construct an iterative process converging to a minimal semiregular solution of the transformed problem for an appropriately chosen starting point. We prove that any non-empty spectrum of the boundary-value problem is a ray [lambda*, +infinity), where lambda* > 0. As an application, we consider the Gol'dshtik mathematical model for separated flows of an incompressible fluid. We show that it satisfies the hypotheses of our theorem and has a non-empty spectrum.