  Unique existence of analytically strong solutions to stochastic partial
differential equations (SPDE) with drift given by the subdifferential of a
quasi-convex function and with general multiplicative noise is proven. The
proof applies a genuinely new method of weighted Galerkin approximations based
on the "distance" defined by the quasi-convex function. Spatial regularization
of the initial condition analogous to the deterministic case is obtained. The
results yield a unified framework which is applied to stochastic generalized
porous media equations, stochastic generalized reaction diffusion equations and
stochastic generalized degenerated p-Laplace equations. In particular, higher
regularity for solutions of such SPDE is obtained.
