Ground states and high energy solutions of the planar Schrödinger–Poisson system. (14th August 2017)

• Record Type: Journal Article
• Authors: Du, Miao; Weth, Tobias
• Is Part Of: Nonlinearity. Volume 30:Number 9(2017:Sep.)
• Page Start: 3492
• Page End: 3515
• Publication Date: 2017-08-14
• Subjects: Schrödinger–Poisson system -- logarithmic convolution potential -- ground state solutions -- variational methods; 35J50 -- 35Q40; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa7eac
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper, we are concerned with the Schrödinger–Poisson system 0.1 { − Δ u + u + ϕ u = | u | p − 2 u i n R d, Δ ϕ = u 2 i n 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 S and Weth T (2016 On the planar Schrödinger–Poisson system Ann. Inst. Henri Poincare 33 169–97) that the variational structure of (001 ) differs substantially in the case d = 2 and leads to a richer structure of the set of solutions. However, the variational approach of Cingolani S and Weth T (2016 On the planar Schrödinger–Poisson system Ann. Inst. Henri Poincare 33 169–97) 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 < p < 4 with a different variational approach.