Spectrum, global bifurcation and nodal solutions to Kirchhoff-type equations
Files
Date
2018-11-05
Authors
Cao, Xiaofei
Dai, Guowei
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
In this article, we consider a Dancer-type unilateral global bifurcation for the Kirchhoff-type problem
-(α + b ∫1 0 |u′|2 dx)u″ = λu + h(x, u, λ) in (0, 1),
u(0) = u(1) = 0.
Under natural hypotheses on h, we show that (αλk, 0) is a bifurcation point of the above problem. As applications we determine the interval of λ, in which there exist nodal solutions for the Kirchhoff-type problem
-(α + b ∫1 0 |u′|2 dx)u″ = λƒ(x, u) in (0, 1),
u(0) = u(1) = 0,
where ƒ is asymptotically linear at zero and is asymptotically 3-linear at infinity. To do this, we also establish a complete characterization of the spectrum of a nonlocal eigenvalue problem.
Description
Keywords
Bifurcation, Spectrum, Nonlocal problem, Nodal solution
Citation
Cao, X., & Dai, G. (2018). Spectrum, global bifurcation and nodal solutions to Kirchhoff-type equations. <i>Electronic Journal of Differential Equations, 2018</i>(179), pp. 1-10.