  This paper introduces a new $p$-dependent coercivity condition through which
$L^p$-moments for solutions can be obtained for a large class of SPDEs in the
variational framework. If $p=2$, our condition reduces to the classically
coercivity condition, which only yields second moments for the solution. The
abstract result is shown to be optimal. Moreover, the results are applied to
obtain $L^p$-moments of solutions for several classical SPDEs such as
stochastic heat equations with Dirichlet and Neumann boundary conditions,
Burgers' equation and the Navier-Stokes equations in two spatial dimensions.
Furthermore, we can recover recent results for systems of SPDEs and higher
order SPDEs using our unifying coercivity condition.
