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Ground State and Sign-Changing Solutions for Critical Schrödinger–Poisson System with Lower Order Perturbation

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Abstract

This article is devoted to study the following Schrödinger–Poisson system

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+K(x)\phi u=a(x)|u|^{p-2}u+|u|^4u, &{}x\in {\mathbb {R}}^3, \\ -\Delta \phi =K(x)u^2, &{}x\in \mathbb {R}^3, \end{array} \right. \end{aligned}$$

where \(4<p<6\), V(x), K(x) and a(x) are nonnegative functions. Under some reasonable conditions on V(x), K(x) and a(x), particularly a(x) can be unbounded, we first investigate the existence of one positive ground state solution and the corresponding energy estimate with the help of Nehari manifold. Meanwhile, its regularity is established through the interior \(L^p\)-estimate. Subsequently, heavily relying on the above results, especially the regularity, we show that the problem possesses one least energy sign-changing solution with precisely two nodal domains by employing constraint variational method and the deformation lemma due to Hofer. Moreover, energy doubling is established, that is, the energy of sign-changing solution is strictly larger than that of the ground state solution.

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The authors declare that data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Authors and Affiliations

  1. School of Mathematical Sciences, TianGong University, Tianjin, 300387, People’s Republic of China

    Ziheng Zhang

  2. Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Ying Wang

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  1. Ziheng Zhang
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Y. Wang wrote the main manuscript text. Z. Zhang reviewed the manuscript.

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Zhang, Z., Wang, Y. Ground State and Sign-Changing Solutions for Critical Schrödinger–Poisson System with Lower Order Perturbation. Qual. Theory Dyn. Syst. 22, 76 (2023). https://doi.org/10.1007/s12346-023-00764-5

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