SOME SINGULAR NONLINEAR BOUNDARY-VALUE-PROBLEMS

Two-point boundary value problems associated with the (possibly) singular nonlinear ordinary differential equation y" + g(x, y') + f(x, y) = 0, a less-than-or-equal-to x less-than-or-equal-to b, are considered. The goal is to obtain rather general existence and uniqueness theorems for positive solutions. In the case of general separated linear boundary conditions, the results allow f(x, y) to be singular as y --> 0+ and at the endpoints, with significant nonlinearity in both f and g. For the special condition y'(a) = 0, the results also allow g to be singular as x --> a+. In this way, the case g(x, y') = ((N - 1)/x)y', which arises when seeking radial solutions of del2y = f(x, y), is included. The results extend previous theorems of Taliaferro and more recent theorems of Gatica, Waltman, et al.