Ground state sign-changing solution for Schrödinger-Poisson system with steep potential well

Jin-Cai Kang; Xiao-Qi Liu; Chun-Lei Tang

doi:10.3934/dcdsb.2022112

Article Contents

2023, Volume 28, Issue 2: 1068-1091. Doi: 10.3934/dcdsb.2022112

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Ground state sign-changing solution for Schrödinger-Poisson system with steep potential well

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

^*Corresponding author: Chun-Lei Tang
^*Corresponding author: Chun-Lei Tang

Received: October 31, 2021

Revised: March 31, 2022

Early access: June 2022

Published: February 2023

This work is supported by National Natural Science Foundation of China (No.11971393)

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Abstract

In the present paper, we investigate a class of nonlinear Schrödinger-Poisson system

$ \begin{equation*} \begin{cases} -\Delta u+V_\lambda(x)u+\mu \phi u = f(u) \quad \quad \ {\rm{in}} \ {\mathbb{R}}^{3},\\ -\Delta \phi = u^2 \quad \quad \quad \quad \ \ \quad \quad \quad \quad \quad \quad {\rm{in}} \ {\mathbb{R}}^{3}, \end{cases} \end{equation*} $

where $ \mu>0 $ and $ V_\lambda(x) = \lambda V(x)+1 $ with $ \lambda>0 $. Under some mild assumptions on $ V $ and $ f $, we prove the existence of ground state sign-changing solution for $ \lambda>0 $ large enough by adopting the deformation lemma and constrained minimization arguments. Then, the least energy of sign-changing solutions is strictly large than two times the ground state energy. Additionally, the phenomenon of concentration for ground state sign-changing solutions is also analysed as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35A01, 35J60.

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