Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials

Rong Cheng; Jun Wang

doi:10.3934/dcdsb.2021317

Article Contents

2022, Volume 27, Issue 11: 6295-6321. Doi: 10.3934/dcdsb.2021317

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Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials

Rong Cheng^1, and
Jun Wang^2, ,

1.
Department of Mathematics, Nanjing University of Information Science & Technology, Nanjing, Jiangsu, 210044, China
2.
Institute of Applied System Analysis, Jiangsu University, Zhenjiang, Jiangsu, 212013, China

^*Corresponding author: Jun Wang
^*Corresponding author: Jun Wang

Received: September 2021

Revised: November 2021

Early access: January 2022

Published: November 2022

This work was supported by NNSF of China(Grants 11971202), Outstanding Young foundation of Jiangsu Province No. BK20200042 and the Six big talent peaks project in Jiangsu Province(XYDXX-015)

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Abstract

In the present paper we study a class of Schrödinger-Poisson equations

$\begin{equation} \begin{cases} -\Delta u+V(x)u+\phi u = a (x)|u|^{p-1}u,\ x\in\mathbb{R}^3\\ -\Delta \phi = u^{2},\ x\in\mathbb{R}^3, \end{cases} \end{equation}\quad\quad\quad (1)$

where $ V(x) $ and $ a(x) $ are of different forms on the half space, i.e. $ V(x) = V_{1}(x), a(x) = a_{1}(x) $ for $ x_{1}>0 $ and $ V(x) = V_{2}(x), a(x) = a_{2}(x) $ for $ x_{1}<0 $, where $ V_{1},V_{2},a_{1} $ and $ a_{2} $ are periodic in each coordinate direction. By using a concentration compactness discussion, we establish the existence of surface gap soliton ground state of (1) for $ p\in [3,5) $. We also give a Mountain-Pass type ground state of (1) for $ p\in (3,5) $.

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Mathematics Subject Classification: Primary: 35J61, 35J20, 35Q55; Secondary: 49J40.

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