Semilinear elliptic Schr\"odinger equations involving singular potentials and source terms

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N>2$) be a $C^2$ bounded domain and $\mathrm{\Sigma }\subset \mathrm{\Omega }$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\le k<N-2$. Put $L_{\mu }=\mathrm{\Delta }+\mu d_{\mathrm{\Sigma }}^{-2}$ in $\mathrm{\Omega }\setminus \mathrm{\Sigma }$, where $d_{\mathrm{\Sigma }}(x)=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\Sigma })$ and $\mu$ is a parameter. We study the boundary value problem (P) $-L_\mu u = g(u)

• \tau$in$\Omega \setminus \Sigma$withcondition$u=\nu$on$\partial \Omega \cup \Sigma$,where$g: $\mathbb{R}$ \to $\mathbb{R}$$ is a nondecreasing, continuous function and $\tau$ and $\nu$ are positive measures. The interplay between the inverse-square potential $d_{\mathrm{\Sigma }}^{-2}$, the nature of the source term $g(u)$ and the measure data $\tau ,\nu$ yields substantial difficulties in the research of the problem. We perform a deep analysis based on delicate estimate on the Green kernel and Martin kernel and fine topologies induced by appropriate capacities to establish various necessary and sufficient conditions for the existence of a solution in different cases.