Positive solutions for a Kirchhoff-Schrödinger-Poisson system with singular term

This work is concerned with a Kirchhoff-Schrödinger-Poisson (KSP) system posed in a bounded domain of ${\mathbb{R}}^3$. The model features a singular nonlinearity $\alpha v^{-\tau }$ with $0<\tau <1$, together with a coupling term of the form $\phi |v{|}^{q-2}v$, where $2<q<3$. The singular term destroys differentiability of the energy functional while the nonlocal potential ${\phi }_v$ causes compactness issues. Using nonsmooth critical point theory, we establish a key estimate linking the weak slope with the derivative of the regular part, prove the Palais-Smale (PS) condition, and characterize critical points as weak solutions. By means of Ekeland's variational principle and the mountain pass theorem, we establish the existence of a constant $\mathrm{\Gamma }>0$ with the property that the system admits two distinct positive solutions whenever $\alpha \in (0,\mathrm{\Gamma })$.