Generalized painlevé equation and superconductivity. asymptotic behavior of unbounded solutions

Consider the nonlinear scalar differential equation 1p(t)(p(t)y′(t))′=−(t)f(t,y(t),p(t)y′(t)), where p and q are positive on (0, 1), “singular” at t = 0. 1 and/or y = 0 and f ϵ C(R+ x R+ x R+, R+), associated to the boundary conditions x(0)=a⩾0, limt→∞ p(t)x′(t)=b⩾0. We prove the existence of a global, positive and strictly increasing solution x = x(t) of this BVP, such that its “derivative” y = p(t)x(t) is also a positive and strictly decreasing map, under a natural growth in f of “superlinear” type. Our approach is based on the analysis of the corresponding vector field on the face-plane (x, px′) and the well-known Knesserʹs type (continuum) technique. As an application, we study the generalized Painlevé equation in a semi-infinite interval (0, +∞) x″= x2n+1 − (t−c)2k+1x2(n−k)−1 which, in turn, models a superheating field attached to a semi-infinite superconductor. Namely, we prove the existence of a (global) strictly positive solution satisfying x′(0)=0, limt→∞x(t)tβ = M, andlimt→∞ tβx′ (t) = βM, where 0 < β < 1, M > 0, and n, k ϵ N with k < n are arbitrary constants.