  In this paper, we consider the backward stochastic differential equation
(BSDE) with generator $f(y)|z|^2,$ where the function $f$ is defined on an open
interval $D$ and locally integrable. The existence and uniqueness of bounded
solutions and $L^p(p\geq1)$ solutions of such BSDEs are obtained. Some
comparison theorems and a converse comparison theorem of such BSDEs are
established. As an application, we give a probabilistic interpretation of
viscosity solution of quadratic PDEs.
