Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable

• Main Author: Guowei Dai
• Format: Article
• Language: English
• Published: University of Szeged 2013-11-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: interval bifurcation; sturm-liouville problem; unilateral global bifurcation
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=2637
• ISSN: 1417-3875; 1417-3875

In this paper, we study unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity for nonlinear Sturm--Liouville problems of the form \begin{equation} \left\{ \begin{array}{l} -\left(pu'\right)'+qu=\lambda au+af\left(x,u,u',\lambda\right)+g\left(x,u,u',\lambda\right)\,\,\text{for}\,\, x\in(0,1),\\ b_0u(0)+c_0u'(0)=0,\\ b_1u(1)+c_1u'(1)=0, \end{array} \right.\nonumber \end{equation} where $a\in C([0, 1], [0,+\infty))$ and $a(x)\not\equiv 0$ on any subinterval of $[0, 1]$, $f,g\in C([0,1]\times\mathbb{R}^3,\mathbb{R})$. Suppose that $f$ and $g$ satisfy \begin{equation} \vert f(x,\xi,\eta,\lambda)\vert\leq M_0\vert \xi\vert+M_1\vert \eta\vert,\,\, \forall x\in [0,1]\,\,\text{and}\,\,\lambda \in\mathbb{R}, \nonumber \end{equation} \begin{equation} g(x,\xi,\eta,\lambda)=o(\vert \xi\vert+\vert \eta\vert),\,\, \text{uniformly in}\,\, x\in [0,1]\,\,\text{and}\,\,\lambda \in \Lambda,\nonumber \end{equation} as either $\vert \xi\vert+\vert \eta\vert\rightarrow 0$ or $\vert \xi\vert+\vert \eta\vert\rightarrow +\infty$, for some constants $M_0$, $M_1$, and any bounded interval $\Lambda$.