Variational solutions of stochastic partial differential equations with cylindrical L\'evy noise

In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation $$\text{dX}(t)=F(X(t))\text{dt}+G(X(t))\text{dL}(t)$$ driven by a cylindrical L'evy process $L$ is established. The coefficients $F$ and $G$ are assumed to satisfy the usual monotonicity and coercivity conditions. The noise is modelled by a cylindrical L'evy processes which is assumed to belong to a certain subclass of cylindrical L'evy processes and may not have finite moments.