A Variational ODE and its Application to an Elliptic Problem

Abstract   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 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. Supported by the National Natural Science Foundation of China (No.10571174, No.10631030) and CAS: KJCX3- SYW-S03.