  We establish a large deviation principle (LDP) for a class of stochastic
porous media equations driven by L\'{e}vy-type noise on a $\sigma$-finite
measure space $(E,\mathcal{B}(E),\mu)$, with the Laplacian replaced by a
negative definite self-adjoint operator. One of the main contributions of this
paper is that we do not assume the compactness of embeddings in the
corresponding Gelfand triple, and to compensate for this generalization, a new
procedure is provided. This is the first paper to deal with LDPs for stochastic
evolution equations with L\'evy noise without compactness conditions. The
coefficient $\Psi$ is assumed to satisfy nondecreasing Lipschitz nonlinearity,
so an important physical problem covered by this case is the Stefan problem.
Numerous examples of negative definite self-adjoint operators are applicable to
our results, for example, for open $E\subset\Bbb{R}^d$, $L=$ Laplacian or
fractional Laplacians, i.e., $L=-(-\Delta)^\alpha,\ \alpha\in(0,1]$,
generalized Schr\"{o}dinger operators, i.e., $L=\Delta+2\frac{\nabla
\rho}{\rho}\cdot\nabla$, Laplacians on fractals is also included.

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