Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics
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 B0 is a parameter, p(0)=0 and p′(t)>0 on (0,∞), f(L)=0 for some L>0 and xf(x)0 if L0xL 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.