Boundary-value problems for shallow elastic membrane caps

Consider the general nonlinear boundary-value problem (p(t)y′(t))′ = p(t)q(t) f(t, y(t), y′(t)), t ≥ 1, g(y(1), y′(1)) = 0, where the function f may be singular at the point y(1) = 0 and p(1) ≥ 0. We obtain conditions which guarantee existence of positive and bounded solutions of the above problem. As an application we prove existence and uniqueness of rotationally symmetric solutions to a nonlinear boundary-value problem, representing the elastic deformation of a spherical cap.