2014/03/23 by Guowei Dai, Dai, Guowei
Engineering · Mathematics · #35B20 #35B32 #35H99 #35R15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1403.5713
openalex publication_date 2014/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \ -(a+b∫Ω\vert ∇ u\vert2 dx)Δu=λu+h(x,u,λ) in Ω,
u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~on Ω. .Under some natural hypotheses on h, we show that (aλ1,0) is a bifurcation point of the above problem. As applications of the above result, we shall determine the interval of λ, in which there exist positive solutions for the above problem with h(x,u;λ)=λf(x,u)-λu, where f is asymptotically linear at zero and is asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we also provide a positive answer to an open problem involving the case of a=0.