Existence and Uniqueness of Constraint Minimizers for the Planar Schrodinger-Poisson System with Logarithmic Potentials

2022/12/01 by Yujin Guo, Guo, Yujin, Wenning Liang +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2212.00234

openalex publication_date 2022/12/01 · openalex created_date 2022/12/13 · openalex updated_date 2026/07/28

Abstract

In this paper, we study constraint minimizers u of the planar Schrödinger-Poisson system with a logarithmic convolution potential ln |x|∗ u² and a logarithmic external potential V(x)=ln (1+|x|²), which can be described by the L²-critical constraint minimization problem with a subcritical perturbation. We prove that there is a threshold ρ^* ∈ (0,∞) such that constraint minimizers exist if and only if 0<ρ<ρ^*. In particular, the local uniqueness of positive constraint minimizers as ρ\nearrowρ^* is analyzed by overcoming the sign-changing property of the logarithmic convolution potential and the non-invariance under translations of the logarithmic external potential.

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• Existence and concentration phenomenon of multiple solutions for the fractional logarithmic Schrödinger-Poisson system via penalization method

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