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Ground State Sign-Changing Solutions for a Schrödinger–Poisson System with Steep Potential Well and Critical Growth

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Abstract

In this article, we consider the existence of ground state sign-changing solutions to a class of Schrödinger–Poisson systems

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V_{\lambda } (x)u+\phi u=|u|^4u+ \mu f(u), &{} ~\textrm{in}~~\mathbb {R}^3, \\ -\Delta \phi =u^2, &{} ~\textrm{in}~~\mathbb {R}^3, \end{array}\right. } \end{aligned}$$

where \(\mu >0\) and \(V_{\lambda }(x)\) = \(\lambda V(x)+1\) with \(\lambda >0\). Under suitable conditions on f and V, by using the constraint variational method and quantitative deformation lemma, we prove that the above problem has one ground state sign-changing solution and the energy of ground state sign-changing solution is strictly more than twice the energy of the ground state solution. Furthermore, we also study the asymptotic behavior of ground state sign-changing solutions as \(\lambda \rightarrow \infty \).

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Acknowledgements

The authors express their gratitude to the reviewers for careful reading and helpful suggestions which led to an improvement of the original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Information, China West Normal University, Nanchong, 637009, Sichuan, People’s Republic of China

    Xiao-Qing Huang & Jia-Feng Liao

  2. College of Mathematics Education, China West Normal University, Nanchong, 637009, Sichuan, People’s Republic of China

    Jia-Feng Liao

  3. Meishan High School, Meishan, 620010, Sichuan, People’s Republic of China

    Rui-Qi Liu

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  1. Xiao-Qing Huang
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Huang Xiao-Qing wrote the main manuscript text, Liao Jia-Feng wrote and revised the manuscript, and Liu Rui-Qi revised the writing of the manuscript.

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Correspondence to Jia-Feng Liao.

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Supported by the Natural Science Foundation of Sichuan(2023NSFSC0073).

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Huang, XQ., Liao, JF. & Liu, RQ. Ground State Sign-Changing Solutions for a Schrödinger–Poisson System with Steep Potential Well and Critical Growth. Qual. Theory Dyn. Syst. 23, 61 (2024). https://doi.org/10.1007/s12346-023-00931-8

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