The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function

Juntao Sun; Tsung-fang Wu

doi:10.3934/dcds.2021011

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2021, Volume 41, Issue 8: 3651-3682. Doi: 10.3934/dcds.2021011

This issue Previous Article Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions Next Article A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo"

The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function

Juntao Sun^1,2, and
Tsung-fang Wu^3, ,

1.
School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China
2.
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
3.
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

^* Corresponding author: Tsung-fang Wu
^* Corresponding author: Tsung-fang Wu

Received: July 31, 2019

Revised: October 31, 2020

Early access: January 2021

Published: August 2021

J. Sun is supported by the National Natural Science Foundation of China (Grant No. 11671236). T.F. Wu is supported in part by the Ministry of Science and Technology, Taiwan (Grant 108-2115-M-390-007-MY2)

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Abstract

In this paper, we study the multiplicity of two spikes nodal solutions for a nonautonomous Schrödinger–Poisson system with the nonlinearity $ f(x)\vert u\vert ^{p-2}u(2<p<6) $ in $ \mathbb{R}^{3} $. By assuming that the weight function $ f\in C(\mathbb{R}^{3},\mathbb{R}^{+}) $ has $ m $ maximum points in $ \mathbb{R}^{3} $, we conclude that such system admits $ m^{2} $ distinct nodal solutions, each of which has exactly two nodal domains. The proof is based on a natural constraint approach developed by us as well as the generalized barycenter map.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35J61, 35Q40.

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