Spectrum, global bifurcation and nodal solutions to Kirchhoff-type equations
Texas State University, Department of Mathematics
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.
Bifurcation, Spectrum, Nonlocal problem, Nodal solution
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.