  In this paper, we study the multi-dimensional backward stochastic
differential equations (BSDEs) whose generator depends also on the mean of both
variables. When the generator is diagonally quadratic, we prove that the BSDE
admits a unique local solution with a fixed point argument. When the generator
has a logarithmic growth of the off-diagonal elements (i.e., for each $i$, the
$i$-th component of the generator has a logarithmic growth of the $j$-th row
$z^j$ of the variable $z$ for each $j \neq i$), we give a new apriori estimate
and obtain the existence and uniqueness of the global solution.
