Existence and multiplicity results for a new p(x)-Kirchhoff problem. (January 2020)

• Record Type: Journal Article
• Authors: Hamdani, Mohamed Karim; Harrabi, Abdellaziz; Mtiri, Foued; Repovš, Dušan D.
• Is Part Of: Nonlinear analysis. Volume 190(2020)
• Publication Date: 2020-01
• Subjects: Variable exponent -- Nonlocal Kirchhoff equation -- p(x)-Laplacian operator -- Palais–Smale condition -- Mountain Pass theorem -- Fountain theorem; 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.2019.111598
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this work, we study the existence and multiplicity results for the following nonlocal p ( x ) -Kirchhoff problem: (0.1) − a − b ∫ Ω 1 p ( x ) | ∇ u | p ( x ) d x d i v ( | ∇ u | p ( x ) − 2 ∇ u ) = λ | u | p ( x ) − 2 u + g ( x, u ) in Ω, u = 0, on ∂ Ω, where a ≥ b > 0 are constants, Ω ⊂ R N is a bounded smooth domain, p ∈ C ( Ω ¯ ) with N > p ( x ) > 1, λ is a real parameter and g is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.