Nonsymmetric solutions of a nonlinear boundary value problem

We study the existence and multiplicity of positive nonsymmetric and signchanging nonantisymmetric solutions of a nonlinear second order ordinary differential equation with symmetric nonlinear boundary conditions, where both of the nonlinearities are of power type. The given problem has already been studied by other authors, but the number of its positive nonsymmetric and sign-changing nonantisymmetric solutions has been determined only under some technical conditions. It was a long-standing open question whether or not these conditions can be omitted. In this article we provide the answer. Our main tool is the shooting method.