A semilinear system of Schrödinger–Maxwell equations. (May 2020)

• Record Type: Journal Article
• Authors: Boccardo, Lucio; Orsina, Luigi
• Is Part Of: Nonlinear analysis. Volume 194(2020)
• Publication Date: 2020-05
• Subjects: Saddle points -- Schrödinger–Maxwell equations -- Regularizing effect; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.02.007
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we are going to prove existence and regularity results for positive solutions of the following elliptic system: − div ( M ( x ) ∇ u ) + r φ u r − 1 = f + φ r, − div ( M ( x ) ∇ φ ) + r u φ r − 1 = u r . where Ω is a bounded open subset of R N, M is a bounded, uniformly elliptic matrix, r > 1, and f ≥ 0 belongs to some Lebesgue space L m ( Ω ), with m ≥ 1 . We will also prove the relationships of the solutions of the system with saddle points of the integral functional J ( v, ψ ) = 1 2 ∫ Ω M ( x ) ∇ v ⋅ ∇ v − 1 2 ∫ Ω M ( x ) ∇ ψ ⋅ ∇ ψ + ∫ Ω | v | r ψ − ∫ Ω | ψ | r v − ∫ Ω f v