© 2001 by London Mathematical Society
© The London Mathematical Society
Multiple Positive Solutions of Semilinear Differential Equations with Singularities
Department of Mathematics and Statistics, York University Toronto, Ontario, Canada M3J 1P3
Received 27 March 2000. Revision received 30 August 2000.
The existence of positive solutions of a second order differential equation of the form
z''+g(t)f(z)=0 (1.1)
with the separated boundary conditions: 
z(0) – ßz'(0) = 0 and 
z(1)+
z'(1) = 0 has proved to be important in physics and applied mathematics. For example, the Thomas–Fermi equation, where f = z3/2 and g = t–1/2 (see [12, 13, 24]), so g has a singularity at 0, was developed in studies of atomic structures (see for example, [24]) and atomic calculations [6]. The separated boundary conditions are obtained from the usual Thomas–Fermi boundary conditions by a change of variable and a normalization (see [22, 24]). The generalized Emden–Fowler equation, where f = zp, p > 0 and g is continuous (see [24, 28]) arises in the fields of gas dynamics, nuclear physics, chemically reacting systems [28] and in the study of multipole toroidal plasmas [4]. In most of these applications, the physical interest lies in the existence and uniqueness of positive solutions.