By using general means, some oscillation criteria for second order nonlinear elliptic differential equation with damping $$sum_{i,j=1}^{N}D_i[a_{ij}(x)D_iy]+sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0$$ are obtained. These criteria are of a high degree of generality and extend the oscillation theorems for second order linear ordinary differential equations due to Kamenev, Philos and Wong.