Oscillatory solutions of singular equations arising in hydrodynamics

Abstract

We investigate the singular differential equation (p(t)u′(t))′=p(t)f(u(t)) on the half-line [0,∞), where f satisfies the local Lipschitz condition on ℝ and has at least two simple zeros. The function p is continuous on [0,∞) and has a positive continuous derivative on (0,∞) and p(0)=0. We bring additional conditions for f and p under which the equation has oscillatory solutions with decreasing amplitudes.