Existence and asymptotic behavior for ground state sign-changing solutions of fractional Schrödinger-Poisson system with steep potential well

Xiao Qing Huang; Jia Feng Liao; Xiao Qing Huang; Jia Feng Liao

doi:10.3934/cam.2024015

Communications in Analysis and Mechanics

2024, Volume 16, Issue 2: 307-333. doi: 10.3934/cam.2024015

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Research article

Existence and asymptotic behavior for ground state sign-changing solutions of fractional Schrödinger-Poisson system with steep potential well

Xiao Qing Huang ^{1
,},
Jia Feng Liao ^{1,2
,
,}

1.
School of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637009, People's Republic of China
2.
College of Mathematics Education, China West Normal University, Nanchong, Sichuan 637009, People's Republic of China

Received: 16 October 2023 Revised: 30 January 2024 Accepted: 13 March 2024 Published: 11 April 2024
35A15; 35B40; 35J20; 35J60

In this paper, we investigate the existence of ground state sign-changing solutions for the following fractional Schrödinger-Poisson system

$ \begin{equation} \begin{cases} (-\Delta)^s u+V_{\lambda} (x)u+\mu\phi u = f(u), & \; \mathrm{in}\; \; \mathbb{R}^3, \\ (-\Delta)^t \phi = u^2, & \; \mathrm{in}\; \; \mathbb{R}^3, \end{cases} \nonumber \end{equation} $

where $ \mu > 0, s\in(\frac{3}{4}, 1), t\in(0, 1) $ and $ V_{\lambda}(x) $ = $ \lambda V(x)+1 $ with $ \lambda > 0 $. Under suitable conditions on $ f $ and $ V $, by using the constraint variational method and quantitative deformation lemma, if $ \lambda > 0 $ is large enough, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $ \mu > 0 $, the least energy of the sign-changing solution is strictly more than twice of the energy of the ground state solution. In addition, we discuss the asymptotic behavior of ground state sign-changing solutions as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.
- fractional Schrödinger-Poisson system,
- variational method,
- sign-changing solution,
- steep potential well
Citation: Xiao Qing Huang, Jia Feng Liao. Existence and asymptotic behavior for ground state sign-changing solutions of fractional Schrödinger-Poisson system with steep potential well[J]. Communications in Analysis and Mechanics, 2024, 16(2): 307-333. doi: 10.3934/cam.2024015

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Abstract

In this paper, we investigate the existence of ground state sign-changing solutions for the following fractional Schrödinger-Poisson system

$ \begin{equation} \begin{cases} (-\Delta)^s u+V_{\lambda} (x)u+\mu\phi u = f(u), & \; \mathrm{in}\; \; \mathbb{R}^3, \\ (-\Delta)^t \phi = u^2, & \; \mathrm{in}\; \; \mathbb{R}^3, \end{cases} \nonumber \end{equation} $

where $ \mu > 0, s\in(\frac{3}{4}, 1), t\in(0, 1) $ and $ V_{\lambda}(x) $ = $ \lambda V(x)+1 $ with $ \lambda > 0 $. Under suitable conditions on $ f $ and $ V $, by using the constraint variational method and quantitative deformation lemma, if $ \lambda > 0 $ is large enough, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $ \mu > 0 $, the least energy of the sign-changing solution is strictly more than twice of the energy of the ground state solution. In addition, we discuss the asymptotic behavior of ground state sign-changing solutions as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.

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