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

Abstract

In this paper the even-order quasilinear ordinary differential equation \( D(\alpha _{n})D(\alpha _{n-1})\cdots D(\alpha _{1})x + p(t)|x|^{\beta -1}x=0 \)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 \( x^{(n)} + p(t)|x|^{\beta -1}x=0 \)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.