Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

Caixia Chen; Aixia Qian

doi:10.3934/mfc.2021036

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2022, Volume 5, Issue 2: 113-128. Doi: 10.3934/mfc.2021036

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Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

Caixia Chen^1, , and
Aixia Qian^2,

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
^* Corresponding author: Caixia Chen

Received: July 2021

Early access: December 2021

Published: May 2022

Supported by the Shandong Province Science Foundation ZR2021MA096 and ZR2020MA005.

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J05, 35J20; Secondary: 35J60.

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