  In this paper, we establish large deviation principle for the strong solution
of a doubly nonlinear PDE driven by small multiplicative Brownian noise.
Motononicity arguments and the weak convergence approach have been exploited in
the proof. Moreover, by using certain a-priori estimates and sequentially
weakly Feller property of the associated Markov semigroup, we show existence of
invariant probability measure for the strong solution of the underlying
problem.
