Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $  \mathbb{R} ^{3} $

Lun Guo; Wentao Huang; Huifang Jia

doi:10.3934/cpaa.2019079

Article Contents

2019, Volume 18, Issue 4: 1663-1693. Doi: 10.3934/cpaa.2019079

This issue Previous Article New general decay results in a finite-memory bresse system Next Article Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium

Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $

1.
College of Science, Huazhong Agricultural University, Wuhan, 430070, China
2.
School of Science, East China JiaoTong University, Nanchang, 330013, China
3.
School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

^* Corresponding author
^* Corresponding author

Received: May 2018

Revised: October 2018

Published: January 2019

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Abstract

We consider the existence of positive solutions for the following fractional Schrödinger-Poisson system

$ \begin{equation*} \begin{cases} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u+\phi(x)u = K(x)f(u)+|u|^{2_{s}^{*}-2}u, \ \ & x\in \mathbb{R} ^3, \\ \varepsilon^{2s}(-\Delta)^{s}\phi = u^{2}, \ \ & x \in \mathbb{R} ^3, \end{cases} \end{equation*} $

where $ s \in (\frac{3}{4}, 1) $, $ \varepsilon $ is a small and positive parameter, $ V $ and $ K $ are nonnegative potential functions. $ 2_{s}^{*} $ is the critical exponent with respect to fractional Sobolev embedding theorem. Under some suitable conditions on the nonlinearity $ f $ and potential functions $ V $ and $ K $, we prove that for $ \varepsilon $ small, the system has a positive ground state solution concentrating around a concrete set related to $ V $ and $ K $. This result generalizes the result for fractional Schrödinger-Poisson system with subcritical exponent by Yu et al. [39] to critical exponent. Moreover, when $ V $ attains its minimum and $ K $ attains its maximum, we also obtain multiple solutions by Ljusternik-Schnirelmann theory.

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Mathematics Subject Classification: Primary: 35J62; Secondary: 35J20, 35J60.

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