  Under a generalized Mokobodzki condition for reflected BSDEs with two
continuous barriers which relates the growth of the generator $g$ and that of
the barriers, we establish several existence and uniqueness results on $L^p\
(p>1)$ solutions of doubly reflected BSDEs with generators satisfying a
one-sided Osgood condition together with a general growth in the state variable
$y$, and a uniform continuity condition or a linear growth condition in the
state variable $z$. This Mokobodzki condition is also proved to be necessary
for existence of the $L^p$ solutions. And, we prove that the $L^p$ solutions
can be approximated by the penalization method and by some sequences of the
$L^p$ solutions of doubly reflected BSDEs.
