Multiplicity of solutions to the generalized extensible beam equations with critical growth. (August 2020)

• Record Type: Journal Article
• Authors: Liang, Sihua; Liu, Zeyi; Pu, Hongling
• Is Part Of: Nonlinear analysis. Volume 197(2020)
• Publication Date: 2020-08
• Subjects: 35J20 -- 35J65 -- 35J60; Fourth-order elliptic equations -- Critical nonlinearity -- Concentration-compactness principle -- Variational method; 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.2020.111835
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper by variational methods, we study the multiplicity of solutions to the following fourth-order elliptic equations of Kirchhoff type with critical nonlinearity in R N : Δ 2 u − M ∫ R N | ∇ u | 2 d x Δ u + V ( x ) u = k ( x ) | u | q − 2 u + λ | u | 2 ∗ ∗ − 2 u, x ∈ R N, where Δ 2 u = Δ ( Δ u ) is the biharmonic operator, M : R + → R + is a continuous function, λ > 0 is a parameter, V : R N → R + is a potential function, and k ( x ) is a nonnegative continuous real valued function satisfying some conditions. The compactness condition is proved by the Lions' second Concentration-compactness principle and Concentration-compactness principle at infinity.