On a class of planar Schrödinger-Poisson system with a bounded potential well

2023/12/12 by Miao Du, Jiaxin Xu, Du, Miao +1
Computer Science · Mathematics · #35J50 #35Q40 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2312.07265

openalex publication_date 2023/12/12 · openalex created_date 2023/12/14 · openalex updated_date 2026/07/28

Abstract

In this paper, we deal with the planar Schrödinger-Poisson system \begincases -Δu + V(x) u + ϕu = b|u|^p-2 u amp;in ℝ²,
Δϕ= u² amp;in ℝ²,\endcases where b ≥ 0, p > 2 and V ∈ C(ℝ², ℝ) is a potential function with inf_ℝ² V >0. Suppose moreover that V exhibits a bounded potential well in the sense that lim_{|x|→ ∞} V(x) exists and is equal to sup_ℝ² V. By using variational methods, we obtain the existence of ground state solutions for this system in the case where p ≥ 3. Furthermore, we also present a minimax characterization of ground state solutions. The main feature of this work is that we do not assume any periodicity or symmetry condition on the external potential V, which is essential to establish the compactness condition of Cerami sequences.

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