Existence and uniqueness of solutions on bounded domains to a FitzHugh-Nagumo type elliptic system

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a transformation to a quasimonotone system. For the transformed system we may and will use maximum (sweeping) principle arguments to derive pointwise estimates. A degree argument completes the uniqueness proof