Differential Equations With Varying Nonlinearity and Singular Perturbations

The purpose of this paper is to contribute to the study of nonlinear differential equations by focusing on specialized families of differential equations in which a nonlinearity varies and grows. For example, the ordinary differential equation y '' +t -2 y ' +y p =0 is the Lane-Emden equation which is of considerable importance in astrophysics. The index p is an index of polytropy and can assume any positive value. The following is a singularly perturbed nonlinear ordinary differential equation with two parameters of varying nonlinearity, p and q. ε 2 y '' +A α (t)y p +B β (t)y q =0· As the varying nonlinearity p grows, intriguing phenomena, reminiscent of singularly perturbed problems, take place. The natural link with the “delta method”and with singularly perturbed problems is discussed. The analysis of certain initial and boundary value problems is given.