Positive bound states for fractional Schrödinger-Poisson system with critical exponent

Xia Sun; Kaimin Teng

doi:10.3934/cpaa.2020165

Article Contents

2020, Volume 19, Issue 7: 3735-3768. Doi: 10.3934/cpaa.2020165

This issue Previous Article Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders Next Article Blow-up for two-component Camassa-Holm equation with generalized weak dissipation

Positive bound states for fractional Schrödinger-Poisson system with critical exponent

Xia Sun and
Kaimin Teng^,

Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

^*Corresponding author
^*Corresponding author

Received: August 31, 2019

Revised: December 31, 2019

Published: April 2020

The work is supported by NSFC grant 11501403, by fund program for the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province (2018) and by the Natural Science Foundation of Shanxi Province (No. 201901D111085)

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Abstract

In this paper, we consider the following fractional critical Schrödinger-Poisson system without perturbation terms

$ \begin{equation*} \begin{cases} (-\Delta)^su+V(x)u+K(x)\phi u = |u|^{2_s^{\ast}-2}u & \hbox{in $\mathbb{R}^3$,} \\ (-\Delta)^t\phi = K(x)u^2& \hbox{in $\mathbb{R}^3$,} \end{cases} \end{equation*} $

where $ s\in(\frac{3}{4},1) $, $ t\in(0,1) $. Under some suitable assumptions on potentials $ V(x) $ and $ K(x) $, we show that the ground state positive solutions do not exist, by using the topological argument, we establish the existence of positive bound state solutions in the range $ (\frac{s}{3}\mathcal{S}_s^{\frac{3}{2s}},\frac{2s}{3}\mathcal{S}_s^{\frac{3}{2s}}) $.

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Mathematics Subject Classification: Primary: 35B38, 35R11; Secondary: 53C35.

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