2015/07/20 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 #Spectral Theory in Mathematical Physics #math.AP
paper · pdf · doi:10.48550/arxiv.1507.05392
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arxiv created 2015/07/20 · openalex publication_date 2015/07/20 · arxiv updated 2015/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the following Kirchhoff type problem \\aligned -(a+b∫_\mathbbBR|∇ u|2dx)Δu= λuq-1 + μup-1, in\mathbbBR,
u>0,\quadin\mathbbBR,
u=0,\quadon∂\mathbbBR, \endaligned .\eqno(P) where \mathbbBR⊂ \bbrN(N≥3) is a ball, 2≤ q<p≤2^*:=(2N)/(N-2) and a, b, λ, μ are positive parameters. By introducing some new ideas and using the well-known results of the problem (P) in the cases of a=μ=1 and b=0, we obtain some special kinds of solutions to (P) for all N≥3 with precise expressions on the parameters a, b, λ, μ, which reveals some new phenomenons of the solutions to the problem (P). It is also worth to point out that it seems to be the first time that the solutions of (P) can be expressed precisely on the parameters a, b, λ, μ, and our results in dimension four also give a partial answer to Neimen's open problems [J. Differential Equations, 257 (2014), 1168--1193]. Furthermore, our results in dimension four seems to be almost "optimal".