Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space

Hangzhou Hu; Yuan Li; Dun Zhao

doi:10.3934/dcdss.2021064

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2021, Volume 14, Issue 6: 1899-1916. Doi: 10.3934/dcdss.2021064

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Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space

School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

^* Corresponding author: Dun Zhao
^* Corresponding author: Dun Zhao

Received: November 30, 2020

Revised: February 28, 2021

Early access: May 2021

Published: June 2021

This work is supported by NSFC under Grant No. 12075102 and 11971212

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Abstract

We consider the following fractional Schrödinger-Poisson equation with combined nonlinearities

$ \begin{equation*} \begin{cases} (-\Delta)^su+\phi u = |u|^{s^*-2}u+\mu|u|^{p-2}u \,\,\,\rm{in}\ \mathbb{R}^3,\\ -\Delta \phi = 4\pi u^2\ \rm{in}\ \mathbb{R}^3, \end{cases} \end{equation*} $

where $ s\in(\frac{3}{4},1) $, $ \mu>0 $, $ p\in(3,s^*) $ and $ s^* = \frac{6}{3-2s} $. By the perturbation approach we prove the existence of the ground state solution in fractional Coulomb-Sobolev space.

Keywords:

Ground state,
Fractional Schrödinger-Poisson equation,
Coulomb-Sobolev space.

Mathematics Subject Classification: Primary: 35A15, 35Q55, 35R11; Secondary: 35Q40.

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References

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