The oscillation of half-linear differential equations with an oscillatory coefficient

We derive lower bounds for the distance between consecutive zeros of a solution of the second order half-linear differential equation (¦y′ (t) ¦α − 1y′ (t)) ′ + q (t) ¦y (t) ¦α − 1 y (t) = 0, (∗) when q (t): (t0, ∞) → R is locally integrable for some t0 > 0. Then we apply these results to the following equations: (p(t) ¦y′ (t) ¦α − 1y′ (t)) ′ + q (t) ¦y (t) ¦α − 1 y (t) = 0, and ∑i=1N|Dy|n-2Diy+h(|x|)|y|n-2y=0 forx∈Ωt0, where 1. ϵ C ([t0, ∞), (0, ∞)) and ∝∞t0 p (t) −1α dt = ∞ i = t6t6xi, D = (D1, …, DN), Ωt0 = {x ϵ RN: ¦x¦≥ t0} is an exterior domain h ϵ C ([t0, ∞), R) α > 0 is a constant; n > 1 and N > 2 are integers.