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Some oscillation criteria are established for the nonlinear damped elliptic differential equation of second order $$ \sum_{i,\,j=1}^{N}D_i[\,a_{ij}(x)D_jy\,]+\sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0, \tag{E} $$ which are different from most known ones in the sense that they are based on a new weighted function $H(r,s,l)$ defined in the sequel. Both the cases when $D_ib_i(x)$ exists for all $i$ and when it does not exist for some $i$ are considered.