Oscillation theorems for second-order half-linear differential equations

AbstractOscillation criteria for the second-order half-linear differential equation [r(t)|ξ′(t)|α−1 ξ′(t)]′ + p(t)|ξ(t)|α−1ξ(t)=0, t ⩾ t0 are established, where α > 0 is a constant and ∫t∞p(s) exists for t ∈ [t0, ∞). We apply these results to the following equation: ∑i=1NDi(|Du(ξ)|n−2Diu(ξ)) + c(|ξ|)|u(ξ)|n−2u(ξ)= 0, ξ ∈ Ωαwhere Di = ∂∂xi, D = (D1,…, DN), Ωa = x ∈ RN : |x| ≥ a} is an exterior domain, and c ∈ C([a, ∞), R), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant