Positive solutions of a second-order singular ordinary differential equation

We study the existence of positive solutions for the second-order ordinary differential equationu″+ku′t=c(t)g(u)with the initial condition u′(0)=0 in bounded intervals [0,M] and in [0,+∞[. Here k>1, c(t) is bounded and the growth of g at infinity is controlled by a critical exponent. In our approach a shooting argument of Beresticky, Lions and Peletier is combined with variational methods.