  In this paper we develop a new weak convergence and compact embedding method
to study the existence and uniqueness of the
$L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes
L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of backward
stochastic differential equations with p-growth coefficients. Then we establish
the probabilistic representation of the weak solution of PDEs with p-growth
coefficients via corresponding BSDEs.
