Existence, multiplicity and non-existence of solutions for modified Schrödinger-Poisson systems

Xian Zhang; Chen Huang; Xian Zhang; Chen Huang

doi:10.3934/mbe.2023163

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 3482-3503. doi: 10.3934/mbe.2023163

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Existence, multiplicity and non-existence of solutions for modified Schrödinger-Poisson systems

Xian Zhang ¹,
Chen Huang ^{2
,
,}

1.
Business school, University of Shanghai for Science and Technology, Shanghai 200093, PR China
2.
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China

Academic Editor: Feliz Manuel Barrão Minhós

Received: 28 August 2022 Revised: 07 November 2022 Accepted: 08 November 2022 Published: 06 December 2022

We consider a class of modified Schrödinger-Poisson systems with general nonlinearity by variational methods. The existence and multiplicity of solutions are obtained. Besides, when $ V(x) = 1 $ and $ f(x, u) = |u|^{p-2}u $, we obtain some existence and non-existence results for the modified Schrödinger-Poisson systems.
- modified Schrödinger-Poisson system,
- Nehari manifold,
- genus,
- Pohozaev identity
Citation: Xian Zhang, Chen Huang. Existence, multiplicity and non-existence of solutions for modified Schrödinger-Poisson systems[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 3482-3503. doi: 10.3934/mbe.2023163

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Abstract

We consider a class of modified Schrödinger-Poisson systems with general nonlinearity by variational methods. The existence and multiplicity of solutions are obtained. Besides, when $ V(x) = 1 $ and $ f(x, u) = |u|^{p-2}u $, we obtain some existence and non-existence results for the modified Schrödinger-Poisson systems.

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