Solutions for gauged nonlinear Schrödinger equations on $ {\mathbb R}^2 $ involving sign-changing potentials

Ziqing Yuan; Jing Zhao; Ziqing Yuan; Jing Zhao

doi:10.3934/math.20241036

AIMS Mathematics

2024, Volume 9, Issue 8: 21337-21355. doi: 10.3934/math.20241036

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Solutions for gauged nonlinear Schrödinger equations on $ {\mathbb R}^2 $ involving sign-changing potentials

Ziqing Yuan ^{1
,
,},
Jing Zhao ²

1.
Department of Mathematics, Shaoyang University, Shaoyang, Hunan 422000, China
2.
Big Data College, Tongren University, Tongren, Guizhou 554300, China

Received: 22 May 2024 Revised: 19 June 2024 Accepted: 25 June 2024 Published: 02 July 2024
MSC : 35J85, 47J30, 49J52

This study focused on establishing the existence and multiplicity of solutions for gauged nonlinear Schrödinger equations set on the plane with sign-changing potentials. Our findings contribute to the extension of recent advancements in this area of research. Initially, we examined scenarios where the potential function $ V $ is lower-bounded and the function space has a compact embedding into Lebesgue spaces. Subsequently, we addressed more complex cases characterized by a sign-changing potential $ V $ and a function space that fails to compactly embed into Lebesgue spaces. The proofs of our results are based on the Trudinger-Moser inequality, the application of variational methods, and the utilization of Morse theory.
- Schrödinger equation,
- variational method,
- sign-changing potentival,
- Morse theory,
- Chern-Simons gauge term
Citation: Ziqing Yuan, Jing Zhao. Solutions for gauged nonlinear Schrödinger equations on $ {\mathbb R}^2 $ involving sign-changing potentials[J]. AIMS Mathematics, 2024, 9(8): 21337-21355. doi: 10.3934/math.20241036

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Abstract

This study focused on establishing the existence and multiplicity of solutions for gauged nonlinear Schrödinger equations set on the plane with sign-changing potentials. Our findings contribute to the extension of recent advancements in this area of research. Initially, we examined scenarios where the potential function $ V $ is lower-bounded and the function space has a compact embedding into Lebesgue spaces. Subsequently, we addressed more complex cases characterized by a sign-changing potential $ V $ and a function space that fails to compactly embed into Lebesgue spaces. The proofs of our results are based on the Trudinger-Moser inequality, the application of variational methods, and the utilization of Morse theory.

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