Uniqueness of L solutions for multidimensional BSDEs and for systems of degenerate parabolic PDEs with superlinear growth generator

We deal with the unique solvability of multidimensional backward stochastic differential equations (BSDEs) with a p-integrable terminal condition (p > 1) and a superlinear growth generator. We introduce a new local condition, on the generator (see Assumption (H4)), then we show that it ensures the existence and uniqueness, as well as the L-stability of solutions. Since the generator is of super linear growth, the uniform continuity is then not satisfied. Furthermore, the (local) monotony condition in the y-variable as well as the (local) Lipschitz condition in the z-variable are not needed. Since the assumptions we impose on the coefficient are local in the three variables y, z and ω, we then also cover the BSDEs with stochastic Lipschitz and/or stochastic monotone coefficient. Although we are focused on the multidimensional BSDEs, our uniqueness and stability results are new even in one-dimensional case. As application, we establish the existence and uniqueness of Sobolev solutions to systems of (possibly) degenerate semilinear parabolic partial differential equations (PDEs) having a super linear growth nonlinear term and a p-integrable terminal condition (p > 1). We cover certain systems of PDEs arising in physics, and in particular the logarithmic nonlinearity u log(|u|). The proofs we give are rather non-standard. And in particular, we introduce a new method which consists to show by using BSDEs that the uniqueness for a system of non-homogeneous semilinear PDEs can be derived from the uniqueness for the homogeneous PDE satisfied by its associated linear part.