  We prove the large deviation principle for the law of the solutions to a
class of parabolic semilinear stochastic partial differential equations driven
by multiplicative noise, in $C\big([0,T]:L^\rho(D)\big)$, where $D\subset
{\mathbb R}^d$ with $d\geqslant 1$ is a bounded convex domain with smooth
boundary and $\rho$ is any real, positive and large enough number. The equation
has nonlinearities of polynomial growth of any order, the space variable is of
any dimension, and the proof is based on the weak convergence method.
