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

Abstract

<p>In this article, we consider a Dancer-type unilateral global bifurcation for the Kirchhoff-type problem</p> <pre>-(α + b ∫¹₀ |u′|² dx)u″ = λu + h(x, u, λ) in (0, 1),<br> u(0) = u(1) = 0.</pre> <p>Under natural hypotheses on h, we show that (αλ_k, 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</p> <pre>-(α + b ∫¹₀ |u′|² dx)u″ = λƒ(x, u) in (0, 1),<br> u(0) = u(1) = 0,</pre> <p>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.</p>