Multiple Positive Solutions of Semilinear Differential Equations with Singularities

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: α z (0) − β z ′(0) = 0 and γ z (1)+δ z ′(1) = 0 has proved to be important in physics and applied mathematics. For example, the Thomas–Fermi equation, where f = z 3/2 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 = z p , 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.