Properties of the spectrum of an elliptic boundary value problem with a parameter and a discontinuous nonlinearity

摘要

An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter lambda multiplying a discontinuous nonlinearity on the right-hand side of the equation. The nonlinearity is zero for values of the phase variable not exceeding some positive number in absolute value and grows sublinearly at infinity. For homogeneous boundary conditions, it is established that the spectrum sigma of the nonlinear problem under consideration is closed (sigma consists of those parameter values for which the boundary value problem has a nonzero solution). A positive lower bound and an upper bound are obtained for the smallest value of the spectrum, lambda*. The case when the boundary function is positive, while the nonlinearity is zero for nonnegative values of the phase variable and nonpositive for negative values, is also considered. This problem is transformed into a problem with homogeneous boundary conditions. Under the additional assumption that the nonlinearity is equal to the difference of functions that are nondecreasing in the phase variable, it is proved that sigma = [lambda*, +infinity) and that for each lambda is an element of sigma the problem has a nontrivial semiregular solution. If there exists a positive constant M such that the sum of the nonlinearity and Mu is a function which is nondecreasing in the phase variable u, then for any lambda is an element of sigma the boundary value problem has a minimal nontrivial solution u(lambda)(x). The required solution is semiregular, and u(lambda)(x) is a decreasing mapping with respect to lambda on [lambda*, +infinity). Applications of the results to the Gol'dshtik mathematical model for separated flows in an incompressible fluid are considered.