By using averaging techniques, some oscillation criteria for quasilinear elliptic differential equations of second order ${\sum }_{i,j=1}^N{D}_i[A_{ij}{\left(x\right)\left|Dy\right|}^{p-2}{D}_{j}y]+p\left(x\right)f\left(y\right)=0$ are obtained. These results extend and generalize the criteria for linear differential equations due to Kamenev, Philos and Wong.