Existence of solutions, estimates for the differential operator, and a "separating" set in a boundary value problem for a second-order differential equation with a discontinuous nonlinearity

We study the existence of solutions of the Sturm-Liouville problem with a nonlinearity discontinuous with respect to the state variable. By the variational method, we prove theorems on the existence of semiregular and regular solutions, estimates for the differential operator, and properties of a "separating" set for the considered problem. These results are applied to the one-dimensional Gol'dshtik and Lavrent'ev models of separated flows of an incompressible fluid.