Existence and multiplicity results for a class of semilinear elliptic equations. (November 2020)

• Record Type: Journal Article
• Authors: Bobkov, Vladimir; Drábek, Pavel; Hernández, Jesús
• Is Part Of: Nonlinear analysis. Volume 200(2020)
• Publication Date: 2020-11
• Subjects: 35A01 -- 35A02 -- 35B09 -- 35B30 -- 35B38; Existence -- Multiplicity -- Branches -- Positive solutions -- Non-Lipschitz nonlinearities -- Variational methods -- Flat solutions -- Compact support solutions; 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.2020.112017
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the existence and multiplicity of nonnegative solutions, as well as the behavior of corresponding parameter-dependent branches, to the equation − Δ u = ( 1 − u ) u m − λ u n in a bounded domain Ω ⊂ R N endowed with the zero Dirichlet boundary data, where 0 < m ≤ 1 and n > 0 . When λ > 0, the obtained solutions can be seen as steady states of the corresponding reaction–diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination. In addition to the main new results, we formulate a few relevant conjectures.