Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree–Fock theory. (April 2022)

• Record Type: Journal Article
• Authors: Bernini, Federico; Bieganowski, Bartosz; Secchi, Simone
• Is Part Of: Nonlinear analysis. Volume 217(2022)
• Publication Date: 2022-04
• Subjects: 35Q55 -- 35A15 -- 35J20 -- 58E05; Choquard equation -- Fractional operators -- Hartree–Fock theory -- Sign-changing nonlinearities; 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.2021.112738
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We are interested in a general Choquard equation − Δ + m 2 u − m u + V ( x ) u − μ | x | u = ∫ R N F ( y, u ( y ) ) | x − y | N − α d y f ( x, u ) − K ( x ) | u | q − 2 u under suitable assumptions on the bounded potential V and on the nonlinearity f . Our analysis extends recent results by the second and third author on the problem with μ = 0 and pure-power nonlinearity f ( x, u ) = | u | p − 2 u . We show that, under appropriate assumptions on the potential, whether the ground state does exist or not. Finally, we study the asymptotic behaviour of ground states as μ → 0 + .