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Existence of Nontrivial Solutions for Schrödinger–Poisson Systems with Critical Exponent on Bounded Domains

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Abstract

In this paper, we concern with the following Schrödinger–Poisson system:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\phi u = f(x,u)+u^5 ,\quad &{} x\in \Omega ,\\ -\Delta \phi =u^2,\quad &{} x\in \Omega ,\\ u=\phi =0, \quad &{} x \in \partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^{3}\) and \(u^5\) reaches the Sobolev critical exponent since \(2^*=6\) in dimension 3. Under some appropriate assumptions on f, a new result on the existence of nontrivial solutions is obtained via variational methods.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Belal Almuaalemi, Haibo Chen & Sofiane Khoutir

Authors
  1. Belal Almuaalemi
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  2. Haibo Chen
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  3. Sofiane Khoutir
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Corresponding author

Correspondence to Haibo Chen.

Additional information

Communicated by Ahmad Izani Md. Ismail.

This work was supported by Natural Science Foundation of China (NSFC11671403) and Mathematics and Interdisciplinary Science project of CSU.

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Almuaalemi, B., Chen, H. & Khoutir, S. Existence of Nontrivial Solutions for Schrödinger–Poisson Systems with Critical Exponent on Bounded Domains. Bull. Malays. Math. Sci. Soc. 42, 1675–1686 (2019). https://doi.org/10.1007/s40840-017-0570-0

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  • DOI: https://doi.org/10.1007/s40840-017-0570-0

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