A Variational ODE and its Application to an Elliptic Problem

In this paper,we consider the following ODE problem(P)where f ∈ C((0,+∞)×R,R),f(r,s)goes to p(r)and q(r)uniformly in r>0 as s→0 and s→+∞,respectively,0≤p(r)≤q(r)∈ L~∞(0,∞).Moreover,for r>0,f(r,s)is nondecreasing in s≥0.Some existenceand non-existence of positive solutions to problem(P)are proved without assuming that p(r)≡0 and q(r)hasa limit at infinity.Based on these results,we get the existence of positive solutions for an elliptic problem.