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doi: 10.3934/mfc.2021036
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Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

 1 School of Mathematics and Computer Application Technology, Jining university, Shandong 273155, China 2 School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

* Corresponding author: Caixia Chen

Received  July 2021 Early access December 2021

Fund Project: Supported by the Shandong Province Science Foundation ZR2021MA096 and ZR2020MA005

In this paper, we consider the following Schrödinger-Poisson equation
 \left\{\begin{aligned} &-\triangle u + u + \phi u = u^{5}+\lambda g(u), &\hbox{in}\ \ \Omega, \\\ & -\triangle \phi = u^{2}, & \hbox{in}\ \ \Omega, \\\ & u, \phi = 0, & \hbox{on}\ \ \partial\Omega.\end{aligned}\right.
where
 $\Omega$
is a bounded smooth domain in
 $\mathbb{R}^{3}$
,
 $\lambda>0$
and the nonlinear growth of
 $u^{5}$
reaches the Sobolev critical exponent in three spatial dimensions. With the aid of variational methods and the concentration compactness principle, we prove the problem admits at least two positive solutions and one positive ground state solution.
Citation: Caixia Chen, Aixia Qian. Multiple positive solutions for the Schrödinger-Poisson equation with critical growth. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021036
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