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Existence and asymptotic behavior of positive solutions to some logarithmic Schrödinger–Poisson system

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Abstract

In this paper, we consider the following logarithmic Schrödinger–Poisson system

$$\begin{aligned} \left\{ \begin{aligned}&- \Delta u + V(x) u + \lambda K(x)\phi u = f(u) + u \log u^2,&x \in {\mathbb {R}}^{3},\\&- \Delta \phi - \varepsilon ^4 \Delta _4 \phi = \lambda K(x) u^2,&x \in {\mathbb {R}}^{3},\\ \end{aligned} \right. \end{aligned}$$

which has increasingly received interest due to the indefiniteness of the energy functional and fourth-order term in Poisson equation. By using variational method, we prove the existence and multiplicity of positive solutions. Finally, we obtain the asymptotic behavior of positive solutions as \(\varepsilon \rightarrow 0^+\) and \(\lambda \rightarrow 0^+\), respectively.

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant No. 12171014) and Natural Science Foundation of Shandong Province (Grant No. ZR2020MA005). The authors would like to thank the referees for their valuable and constructive suggestions and comments.

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Supported by the NSFC (12171014, ZR2020MA005).

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, People’s Republic of China

    Lichao Cui & Anmin Mao

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  1. Lichao Cui
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  2. Anmin Mao
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All authors wrote and revised this manuscript.

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Correspondence to Anmin Mao.

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Cui, L., Mao, A. Existence and asymptotic behavior of positive solutions to some logarithmic Schrödinger–Poisson system. Z. Angew. Math. Phys. 75, 30 (2024). https://doi.org/10.1007/s00033-023-02170-y

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