The planar Schrödinger–Poisson system with a positive potential*The author is supported by PRIN 2017JPCAPN Qualitative and quantitative aspects of nonlinear PDEs. (13th July 2021)

• Record Type: Journal Article
• Authors: Azzollini, Antonio
• Is Part Of: Nonlinearity. Volume 34:Number 8(2021)
• Page Start: 5799
• Page End: 5820
• Publication Date: 2021-07-13
• Subjects: Schrodinger–Poisson system -- Standing wave solutions -- logarithmic convolution potential; 35J50 -- 35Q40; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac0230
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper we consider the problem − Δ u ± ϕ u + W ′ ( x, u ) = 0 in R 2, Δ ϕ = u 2 in R 2, where W is assumed nonnegative. In dimension three, the problem with the sign + (we call it ( P + ) ) was considered and solved in [22 ], whereas in the same paper it was showed that no nontrivial solution exists if we consider the sign − (say it ( P − ) ). We provide a general existence result for ( P + ) and two examples falling in the case ( P − ) for which there exists at least a nontrivial solution.