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In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation $${\left(r\left(t\right)\Phi \left(u^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(u\left(t\right)\right)=0,$$ where (i) $r,c\in C([t_0,\infty )$, $ℝ:=(-\infty ,\infty ))$ and $r\left(t\right)>0$ on $[t_0,\infty )$ for some $t_0\ge 0$; (ii) $\Phi \left(u\right)={\left|u\right|}^{p-2}u$ for some fixed number $p>1$. We also generalize some results of Hille-Wintner, Leighton and Willet.