Monotone positive solutions for singular boundary value problems

Consider the nonlinear scalar differential equations1p(t)(p(t)y′(t))′+sign(1−α)q(t)f(t,y(t),p(t)y′(t))=0,where α>0, α≠1,p and q are “singular” at t=0,1 and f∈C((0,1)×R+×R−,R−), associated to boundary conditions γy(0)+δ limt→0+ p(t)y′(t)=0, γ>0,limt→1− p(t)y′(t)=α limt→0+ p(t)y′(t).Existence of a monotone positive solutions of this BVP are given, with their slope a priori bounded, under superlinear or sublinear growth in f. The approach is based on the analysis of the corresponding vector field on the face-plane and the well-known shooting technique.