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

We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichletcondition, 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) fornegative (resp, non-negative) values of the phase variable. Let $\widetilde{u}(x)$ be a solution ofthe boundary-value problem with zero right-hand side (the boundary function is assumed to be positive).Putting $v(x)=u(x)-\widetilde{u}(x)$, we reduce the original problem to a problem with homogeneousboundary condition. The spectrum of the transformed problem consists of the values of the parameterfor 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 semiregularsolution of the transformed problem for an appropriately chosen starting point. We prove that any non-emptyspectrum of the boundary-value problem is a ray $[\lambda^*,+\infty)$, where $\lambda^*>0$. As an application,we consider the Gol'dshtik mathematical model for separated flows of an incompressible fluid. We show thatit satisfies the hypotheses of our theorem and has a non-empty spectrum.

elliptic boundary-value problem, problem with parameter, discontinuous non-linearity, iterative process, minimal solution, semiregular solution, spectrum, Gol'dshtik model