Multiple positive solutions for a bi-nonlocal Kirchhoff-Schr$ \ddot{\mathrm{o}} $dinger-Poisson system with critical growth

Guaiqi Tian; Hongmin Suo; Yucheng An; Guaiqi Tian; Hongmin Suo; Yucheng An

doi:10.3934/era.2022228

Electronic Research Archive

2022, Volume 30, Issue 12: 4493-4506. doi: 10.3934/era.2022228

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Multiple positive solutions for a bi-nonlocal Kirchhoff-Schr$ \ddot{\mathrm{o}} $dinger-Poisson system with critical growth

1.
School of Data Sciences and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
2.
School of Sciences, Guizhou University of Engineering Science, Bijie 551700, China

Academic Editor: Chao Ji

Received: 17 August 2022 Revised: 01 October 2022 Accepted: 09 October 2022 Published: 17 October 2022

In this article, we study the following bi-nonlocal Kirchhoff-Schr$ \ddot{\mathrm{o}} $dinger-Poisson system with critical growth:

$ \begin{equation*} \begin{cases} -\left( \int_{\Omega}|\nabla u|^2dx\right)^r\Delta u+\phi u = u^5+\lambda\left( \int_{\Omega}F(x, u)dx\right)^sf(x, u), & \mathrm{in}\ \ \Omega, \\ -\Delta\phi = u^2, u>0, & \mathrm{in}\ \ \Omega, \\ u = \phi = 0, & \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} $

where $ \Omega\subset \mathbb{R}^3 $ is a smooth bounded domain, $ \lambda > 0 $, $ 0\leq r < 1 $, $ 0 < s < \frac{1-r}{3(r+1)} $ and $ f(x, u) $ satisfies some suitable assumptions. By using the concentration compactness principle, the multiplicity of positive solutions for the above system is established.
- Kirchhoff-Schr$ \ddot{\mathrm{o}} $dinger-Poisson systems,
- positive solutions,
- critical growth,
- concentration compactness principle,
- multiplicity
Citation: Guaiqi Tian, Hongmin Suo, Yucheng An. Multiple positive solutions for a bi-nonlocal Kirchhoff-Schr$ \ddot{\mathrm{o}} $dinger-Poisson system with critical growth[J]. Electronic Research Archive, 2022, 30(12): 4493-4506. doi: 10.3934/era.2022228

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Abstract

In this article, we study the following bi-nonlocal Kirchhoff-Schr$ \ddot{\mathrm{o}} $dinger-Poisson system with critical growth:

$ \begin{equation*} \begin{cases} -\left( \int_{\Omega}|\nabla u|^2dx\right)^r\Delta u+\phi u = u^5+\lambda\left( \int_{\Omega}F(x, u)dx\right)^sf(x, u), & \mathrm{in}\ \ \Omega, \\ -\Delta\phi = u^2, u>0, & \mathrm{in}\ \ \Omega, \\ u = \phi = 0, & \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} $

where $ \Omega\subset \mathbb{R}^3 $ is a smooth bounded domain, $ \lambda > 0 $, $ 0\leq r < 1 $, $ 0 < s < \frac{1-r}{3(r+1)} $ and $ f(x, u) $ satisfies some suitable assumptions. By using the concentration compactness principle, the multiplicity of positive solutions for the above system is established.

References

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