Two positive solutions for the Kirchhoff type elliptic problem with critical nonlinearity in high dimension. (September 2019)

• Record Type: Journal Article
• Authors: Naimen, Daisuke; Shibata, Masataka
• Is Part Of: Nonlinear analysis. Volume 186(2019)
• Page Start: 187
• Page End: 208
• Publication Date: 2019-09
• Subjects: Elliptic -- Critical -- Variational method -- Fibering map -- Nehari manifold -- Kirchhoff; 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.2019.02.003
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We investigate the Kirchhoff type elliptic problem with critical nonlinearity; − 1 + α ∫ Ω | ∇ u | 2 d x Δ u = λ u q + u 2 ∗ − 1, u > 0 in Ω, u = 0 on ∂ Ω, where N ≥ 5, Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω, α > 0, λ ∈ R, 2 ∗ = 2 N ∕ ( N − 2 ), and q ∈ [ 1, 2 ∗ − 1 ) . We prove the existence of two solutions of it via the variational method. Since N ≥ 5 and α > 0, the uniqueness assertion for the associated limiting problem may fail. This causes serious difficulties in controlling concentrating Palais–Smale sequences. We overcome these by introducing new techniques. For a mountain pass type solution, we utilize the limit function of the fibering maps of the concentrating Palais–Smale sequence. This tool is based on our careful setting of Nehari type sets. On the other hand, a suitable modification to a concentrating minimizing sequence enables us to obtain a global minimum solution. This is the first work which proves the multiplicity of positive solutions of the Kirchhoff type critical problem in high dimension.