Global bifurcation and nodal solutions for homogeneous Kirchhoff type equations

• Published in: Electronic journal of qualitative theory of differential equations Vol. 2020; no. 29; pp. 1 - 13
• Main Authors: Liu, Fang, Luo, Hua, Dai, Guowei
• Format: Journal Article
• Language: English
• Published: University of Szeged 01.01.2020
• Subjects: bifurcation; nodal solution; nonlocal problem; regularity results; spectrum
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2020.1.29

In this paper, we shall study unilateral global bifurcation phenomenon for the following homogeneous Kirchhoff type problem \begin{equation*} \begin{cases} -\left(\int_0^1 \left\vert u'\right\vert^2\,dx\right)u''=\lambda u^3+h(x,u,\lambda)&\text{in}\,\, (0,1),\\ u(0)=u(1)=0. \end{cases} \end{equation*} As application of bifurcation result, we shall determine the interval of $\lambda$ in which there exist nodal solutions for the following homogeneous Kirchhoff type problem \begin{equation*} \begin{cases} -\left(\int_0^1 \left\vert u'\right\vert^2\,dx\right) u''=\lambda f(x,u)&\text{in}\,\, (0,1),\\ u(0)=u(1)=0, \end{cases} \end{equation*} where $f$ is asymptotically cubic at zero and infinity. To do this, we also establish a complete characterization of the spectrum of a homogeneous nonlocal eigenvalue problem.