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We establish the existence and uniqueness of strong solutions to stochasticporous media equations driven by L\'{e}vy noise on a $\sigma$-finite measurespace $(E,\mathcal{B}(E),\mu)$, and with the Laplacian replaced by a negativedefinite self-adjoint operator. The coefficient $\Psi$ is only assumed tosatisfy the increasing Lipschitz nonlinearity assumption, without therestriction $r\Psi(r)\rightarrow\infty$ as $r\rightarrow\infty$ for$L^2(\mu)$-initial data. We also extend the state space, which avoids thetransience assumption on $L$ or the boundedness of $L^{-1}$ in$L^{r+1}(E,\mathcal{B}(E),\mu)$ for some $r\geq1$. Examples of the negativedefinite self-adjoint operators include fractional powers of the Laplacian,i.e. $L=-(-\Delta)^\alpha,\ \alpha\in(0,1]$, generalized Schr\"{o}dingeroperators, i.e. $L=\Delta+2\frac{\nabla \rho}{\rho}\cdot\nabla$, and Laplacianson fractals.

Stochastic generalized porous media equations driven by L\'{e}vy noise with increasing Lipschitz nonlinearities