  This paper is devoted to proving the small noise asymptotic behaviour,
particularly large deviation principle, for multi-scale stochastic dynamical
systems with fully local monotone coefficients driven by multiplicative noise.
The main techniques are based on a combination of the weak convergence
approach, the time discretization technique and the theory of pseudo-monotone
operator. The main results derived in this paper have broad applicability to
various multi-scale models, where the slow component could be such as
stochastic porous medium equations, stochastic Cahn-Hilliard equations and
stochastic 2D Liquid crystal equations.
