Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent

Yong-Yong Li; Yan-Fang Xue; Chun-Lei Tang

doi:10.3934/cpaa.2019104

Article Contents

2019, Volume 18, Issue 5: 2299-2324. Doi: 10.3934/cpaa.2019104

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Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent

1.
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China
2.
College of Mathematics and Information Sciences, Xin-Yang Normal University, Xinyang, 464000, China

^* Corresponding author
^* Corresponding author

Received: September 30, 2017

Revised: September 30, 2017

Published: March 2019

The research is supported by National Natural Science Foundation of China (No.11471267)

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Abstract

In this paper, we study the modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system:

$ \begin{align*} \begin{cases} -\Delta u+V(x)u-K(x)\phi|u|^8u-\Delta(u^2)u = g(x,u),\ \ \ \ &\mbox{in}\ \mathbb{R}^3,\\ -\Delta\phi = K(x)|u|^{10},\ \ \ \ &\mbox{in}\ \mathbb{R}^3, \end{cases} \end{align*} $

where $ V,K,g $ are asymptotically periodic functions of $ x $. Based on variational methods and the dual approach, we prove the existence of ground state solution by using the Nehari manifold method, the Mountain Pass theorem and the concentration-compactness principle.

Keywords:

Modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system,
asymptotically periodic,
critical nonlocal term,
Nehari manifold,
ground state solution.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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