Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics

Abstract

This paper investigates singular initial problems ( p ( t ) u ′ ) ′ = p ( t ) f ( u ) , u ( 0 ) = B , u ′ ( 0 ) = 0 , on the half-line [ 0 , ∞ ) . Here B < 0 is a parameter, p ( 0 ) = 0 and p ′ ( t ) > 0 on ( 0 , ∞ ) , f ( L ) = 0 for some L > 0 and x f ( x ) < 0 if L 0 < x < L and x ≠ 0 . The existence of a strictly increasing solution to the problem for which there exists a finite c > 0 such that u ( c ) = L is discussed. This is fundamental for the existence of a strictly increasing solution of the problem having its limit equal to L as t → ∞ , which has great importance in applications.