An elliptic problem with strongly nonlinear interface condition

We consider a system of partial differential equations arising in the modelling of the electrochemical processes in Li-ion batteries. In this simplification, it consists of two elliptic equations in two domains separated by an interface. At the interface holds a Neumann condition that depends exponentially on the jump of the solution at the interface (Butler–Vollmer equation). We prove unique existence of a solution and a comparison principle.