= Compact L2 potential perturbation of the Dirichlet Laplacian
{title2=$K=M_w\Delta_D^{-1},\quad w\in L^p,\ p>n\geq3$}
On a bounded open set, let $Tv\in W_0^{1,2}$ be the <weak solution> of $\Delta Tv=v$. The energy estimate bounds $T:L^2\to W_0^{1,2}$. For $p>n$, put $q=2p/(p-2)<2n/(n-2)$. The zero-boundary <Rellich-Kondrachov compactness theorem> gives compact embedding into $L^q$, and <Hölder's inequality> gives $\|wz\|_2\leq\|w\|_p\|z\|_q$. Hence $Kv=wTv$ is compact on $L^2$, without a regularity assumption on the boundary. Writing $v=\Delta u$ makes $\Delta u-wu=f$ equivalent to $(I-K)v=f$. The <Fredholm alternative for a compact operator> then identifies unique solvability with triviality of the homogeneous kernel.
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