  We consider partial differential equations (PDEs) characterized by an upper
barrier that depends on the solution itself and a fixed lower barrier, while
accommodating a non-local driver. First, we show a Feynman-Kac representation
for the PDE when the driver is local. Specifically, we relate the non-linear
Snell envelope for an optimal stopping problem, where the underlying process is
the first component in the solution to a stopped backward stochastic
differential equation (BSDE) with jumps and a constraint on the jumps process,
to a viscosity solution for the PDE. Leveraging this Feynman-Kac
representation, we subsequently prove existence and uniqueness of viscosity
solutions in the non-local setting by employing a contraction argument. In
addition, the contraction argument yields existence of a new type of non-linear
Snell envelope and extends the theory of probabilistic representation for PDEs.
