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Consider stochastic partial differential equations (SPDEs) with fully localmonotone coefficients in a Gelfand triple $V\subseteq H\subseteq V^*$ $$  \left\{  \begin{align}  &dX_t=A(t,X_t)dt+B(t,X_t)dW_t,\ t\in (0,T]\\\\&  X_0=x\in H,  \end{align}  \right. $$  where  $$A: [0,T] \times V\rightarrow V^*,\ \ B:[0,T]\times V\rightarrow\ L_2(U,H)$$are measurable maps, $L_2(U,H)$ is the space of Hilbert-Schmidt operators from$U$ to $H$ and $W$ is a $U$-cylindrical Wiener process.\par  In this paper, we establish a small noise large deviation principle(LDP) forthe solutions {$u^\varepsilon$}$_{\varepsilon>0}$ of the above SPDEs.  The main contribution of this paper is the much more generality of ourframework than that of the existing results. In particular, the diffusioncoefficient $B(t,\cdot)$ may depend on the gradient of the solutions, which isof great interest in the field of SPDEs, but there are few existing results onthe topic of LDP. The broader scope of the fully local monotone setting leadsus to use different strategies and techniques. A combination of thepseudomonotone technique and compactness arguement plays a crucial role in thewhole paper.  Our framework is very general to include many interesting models that couldnot be covered by existing work, including stochastic quasilinear SPDEs,stochastic convection diffusion equation, stochastic 2D Liquid crystalequation, stochastic $p$-Laplace equation with gradient-dependent noise,stochastic 2D Navier-Stokes equation with gradient-dependent noise etc.

Large deviations of fully local monotone stochastic partial differential equations driven by gradient-dependent noise