We will describe a recent approach to treating the validity of comparison principles for weak solutions of scalar second order PDE of the form F(x,u(x),Hu(x)) = 0 under non standard structural conditions on F. In particular, F(x,r,A) need not be globally monotone in r and A. One exploits Krylov’s notion [Trans AMS ‘95] of elliptic branches to replace the PDE with a differential inclusion involving a certain set-valued map. A natural notion of admissibile viscosity solution for the differential inclusion can be captured in terms of subaffine functions and the notion of duality of Harvey and Lawson [Comm. Pure Appl. Math ‘09]. We will discuss these notions and show how to determine structural conditions on F which ensure that the associated set-valued map is sufficiently regular to yield the desired comparison principle. This is a joint work with Marco Cirant (Università di Milano).