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

Ying Zhou; Wei Wei; Jun Lei; Yue Wang; Ying Zhou; Wei Wei; Jun Lei; Yue Wang

doi:10.3934/era.2026179

Electronic Research Archive

2026, Volume 34, Issue 6: 3991-4004. doi: 10.3934/era.2026179

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Theory article

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

1.
School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China
2.
Guizhou Key Laboratory of Artificial Intelligence and Brain-inspired Computing, Guizhou University, Guiyang 550025, China
3.
School of Mathematics and Big Data, Guizhou Education University, Guiyang 550018, China
4.
School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550018, China

Received: 26 March 2026 Revised: 21 April 2026 Accepted: 08 May 2026 Published: 14 May 2026

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 $ \varphi|v|^{q-2}v $, where $ 2 < q < 3 $. The singular term destroys differentiability of the energy functional while the nonlocal potential $ \varphi_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 $ \Gamma > 0 $ with the property that the system admits two distinct positive solutions whenever $ \alpha\in(0, \Gamma) $.
- weak slope,
- Palais-Smale condition,
- nonlocal potential
Citation: Ying Zhou, Wei Wei, Jun Lei, Yue Wang. Positive solutions for a Kirchhoff-Schrödinger-Poisson system with singular term[J]. Electronic Research Archive, 2026, 34(6): 3991-4004. doi: 10.3934/era.2026179

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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 < \tau < 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 $ \varphi_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 $ \Gamma > 0 $ with the property that the system admits two distinct positive solutions whenever $ \alpha\in(0, \Gamma) $.

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