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

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