Abundance of entire solutions to nonlinear elliptic equations by the variational method. (January 2020)

• Record Type: Journal Article
• Authors: Lerman, L.M.; Naryshkin, P.E.; Nazarov, A.I.
• Is Part Of: Nonlinear analysis. Volume 190(2020)
• Publication Date: 2020-01
• Subjects: 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.2019.111590
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study entire bounded solutions to the equation Δ u − u + u 3 = 0 in R 2 . Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in an unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. It is also applicable for more general equations in any dimension.