2016/05/23 by Yisheng Huang, Huang, Yisheng, Zeng Liu +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1605.06906
22 page, 1 figure
arxiv created 2016/05/23 · openalex publication_date 2016/05/23 · arxiv updated 2016/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider the following Kirchhoff problem \\aligned -(a+b∫Ω|∇ u|2dx)Δu= λuq-1 + μu2^*-1, in Ω,
u>0,\quadin Ω,
u=0,\quadon ∂Ω, \endaligned .\eqno(P) where Ω⊂ \bbrN(N≥4) is a bounded domain, 2≤ q<2^*, 2^*=(2N)/(N-2) is the critical Sobolev exponent and a, b, λ, μ are positive parameters. By using the variational method, we obtain some existence and nonexistence results to (P) for all N≥4 with some further conditions on the parameters a, b, λ, μ, which partially improve some known results in the literatures. Furthermore, Our result for N=4 and q>2, together with our previous works \citeHLW15,HLW151, gives an almost positive answer to Neimen's open question [J. Differential Equations, 257 (2014), 1168--1193].