Existence of positive solutions of higher-order quasilinear ordinary differential equations

In this paper the even-order quasilinear ordinary differential equation is considered under the hypotheses that n is even, D(α _i )x = (|x|αi−1 x)′, α _i > 0(i = 1,2,…, n), β > 0, and p(t) is a continuous, nonnegative, and eventually nontrivial function on an infinite interval [a, ∞), a > 0. The existence of positive solutions of (1.1) is discussed, and basic results to the classical equation are extended to the more general equation (1.1). In particular, necessary and sufficient integral conditions for the existence of positive solutions of (1.1) are established in the case α ₁α₂⋅s α _n ≠ β. This research was partially supported by Grant-in-Aid for Scientific Research (No. 15340048), Japan Society for the Promotion of Science. Mathematics Subject Classification (2000) 34C10, 34C11