Some annulus oscillation criteria are established for the second order damped elliptic differential equation $$sumlimits_{i,j=1}^N D_i[a_{ij}(x)D_jy]+sumlimits_{i=1}^Nb_i(x)D_iy+C(x,y)=0$$ under quite general assumption that they are based on the information only on a sequence of annuluses of $Omega(r_0)$ rather than on the whole exterior domain $Omega(r_0)$. Our results are extensions of those due to Kong for ordinary differential equations. In particular, the results obtained here can be applied to the extreme case such as ${int}_{Omega(r0)}c(x)dx=-infty$.