  In this paper we deal with the problem of the existence and the uniqueness of
a solution for one dimensional reflected backward stochastic differential
equations with two strictly separated barriers when the generator is allowing a
logarithmic growth $(|y||\ln|y||+|z|\sqrt{|\ln|z||})$ in the state variables
$y$ and $z$. The terminal value $\xi$ and the obstacle processes $(L_t)_{0\leq
t\leq T}$ and $(U_t)_{0\leq t\leq T}$ are $L^p$-integrable for a suitable $p >
2$. The main idea is to use the concept of local solution to construct the
global one. As applications, we broaden the class of functions for which mixed
zero-sum stochastic differential games admit an optimal strategy and the
related double obstacle partial differential equation problem has a unique
viscosity solution.
