Stochastic generalized porous media equations driven by Lévy noise with increasing Lipschitz nonlinearities

We establish the existence and uniqueness of strong solutions to stochastic porous media equations driven by Lévy noise on a $$\sigma $$ -finite measure space $$(E,{\mathcal {B}}(E),\mu )$$ , with Laplacian replaced negative definite self-adjoint operator. The coefficient $$\Psi is only assumed satisfy increasing Lipschitz nonlinearity assumption, without restriction $$r\Psi (r)\rightarrow \infty as $$r\rightarrow for $$L^2(\mu -initial data. also extend state space, which avoids transience assumption L or boundedness $$L^{-1}$$ in $$L^{r+1}(E,{\mathcal some $$r\ge 1$$ . Examples operators include fractional powers Laplacian, i.e., $$L=-(-\Delta )^\alpha ,\ \alpha \in (0,1]$$ generalized $$\mathrm Schr\ddot{o}dinger$$ operators, $$L=\Delta +2\frac{\nabla \rho }{\rho }\cdot \nabla Laplacians fractals.