Ground states and high energy solutions of the planar Schrödinger-Poisson system

2017/03/15 by Miao Du, Du, Miao, Tobias Weth +1 · 1 citation
Mathematics · #35J50 #35Q40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1703.05090

openalex publication_date 2017/03/15 · openalex created_date 2017/03/23 · openalex updated_date 2026/07/28

Abstract

In this paper, we are concerned with the Schrödinger-Poisson system (0.1) -Δu + u +ϕu = |u|^p-2u in ℝ^d, Δϕ= u² in ℝ^d. Due to its relevance in physics, the system has been extensively studied and is quite well understood in the case d ≥ 3. In contrast, much less information is available in the planar case d=2 which is the focus of the present paper. It has been observed by Cingolani and the second author \citeCingolani-Weth-2016 that the variational structure of (0.1) differs substantially in the case d=2 and leads to a richer structure of the set of solutions. However, the variational approach of \citeCingolani-Weth-2016 is restricted to the case p ≥ 4 which excludes some physically relevant exponents. In the present paper, we remove this unpleasant restriction and explore the more complicated underlying functional geometry in the case 2

Citations

• Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations

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• Ground states of the planar nonlinear Schrödinger--Newton system with a point interaction

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