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,∞)[0,∞). Here B<0B<0 is a parameter, p(0)=0p(0)=0 and p′(t)>0p′(t)>0 on (0,∞)(0,∞), f(L)=0f(L)=0 for some L>0L>0 and xf(x)<0xf(x)<0 if L00c>0 such that u(c)=Lu(c)=L is discussed. This is fundamental for the existence of a strictly increasing solution of the problem having its limit equal to LL as t→∞t→∞, which has great importance in applications.