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

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Abstract

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 < τ < 1, together with a coupling term of the form $\varphi|v|^{q-2} v$, where 2 < q < 3. The singular term destroys differentiability of the energy functional while the nonlocal potential φ_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 Γ > 0 with the property that the system admits two distinct positive solutions whenever $\alpha \in(0, \Gamma)$.

Keywords

weak slope / Palais-Smale condition / nonlocal potential

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