Spectrum, global bifurcation and nodal solutions to Kirchhoff-type equations

Date

2018-11-05

Authors

Cao, Xiaofei
Dai, Guowei

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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.

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Attribution 4.0 International

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