Nonlinear scalar field equations with general nonlinearity. (January 2020)

• Record Type: Journal Article
• Authors: Jeanjean, Louis; Lu, Sheng-Sen
• Is Part Of: Nonlinear analysis. Volume 190(2020)
• Publication Date: 2020-01
• Subjects: 35J20 -- 35J60; Nonlinear scalar field equations -- Berestycki–Lions nonlinearity -- Nonradial solutions -- Monotonicity trick; 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.2019.111604
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Consider the nonlinear scalar field equation (0.1) − Δ u = f ( u ) in R N, u ∈ H 1 ( R N ), where N ≥ 3 and f satisfies the general Berestycki–Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski (2017) made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski's results on (0.1) . The keys to our approach are an extension to the symmetric mountain pass setting of the monotonicity trick, and a new decomposition result for bounded Palais–Smale sequences.