A variational ODE and its application to an elliptic problem

In this paper, we consider the following ODE problem {-u ''(R) + (N - 1) (N - 3)/4r(2) u(r) + lambda u(r) = f(r, r(1-N/2) u)u(r), r>0, u is an element of H, N >= 3. where f is an element of C((0, +infinity) x R, R), f (r, s) goes to p(r) and q(r) uniformly in r > 0 as s -> 0 and s -> +infinity, respectively, 0 <= p(r) <= q(r) is an element of L-infinity (0, infinity). Moreover, for r > 0, f (r, s) is nondecreasing in s >= 0. Some existence and non-existence of positive solutions to problem (P) are proved without assuming that p(r) 0 and q(r) has a limit at infinity. Based on these results, we get the existence of positive solutions for an elliptic problem.