Two positive solutions for the Kirchhoff type elliptic problem with critical nonlinearity in high dimension

• Published in: Nonlinear analysis 2019-09, Vol.186, p.187-208
• Main Authors: Naimen, Daisuke, Shibata, Masataka
• Format: Article
• Language: English
• Subjects: Critical; Elliptic; Fibering map; Kirchhoff; Nehari manifold; Nonlinearity; Smooth boundaries; Variational method
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2019.02.003

We investigate the Kirchhoff type elliptic problem with critical nonlinearity; −1+α∫Ω|∇u|2dxΔu=λuq+u2∗−1,u>0inΩ,u=0on∂Ω,where N≥5, Ω⊂RN is a bounded domain with smooth boundary ∂Ω, α>0, λ∈R, 2∗=2N∕(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.