  In this paper, we first establish the existence and uniqueness of $L^p\
(p>1)$ solutions for multidimensional backward stochastic differential
equations (BSDEs) under a weak monotonicity condition together with a general
growth condition in $y$ for the generator $g$. Then, we overview several
conditions related closely to the weak monotonicity condition and compare them
in an effective way. Finally, we put forward and prove a stability theorem and
a comparison theorem of $L^p\ (p>1)$ solutions for this kind of BSDEs.
