Global bifurcation result and nodal solutions for Kirchhoff-type equation

Abstract

We investigate the global structure of nodal solutions for the Kirchhoff-type problem $ \left\{\begin{array}{ll} -(a+b\int_{0}^{1}|u'|^2dx)u'' = \lambda f(u),\ x\in (0,1),\\[2ex] u(0) = u(1) = 0, \end{array} \right. $ where $ a > 0, b > 0 $ are real constants, $ \lambda $ is a real parameter. $ f\in C(\mathbb{R}, \mathbb{R}) $ and there exist four constants $ s_1\leq s_2 < 0 < s_3\leq s_4 $ such that $ f(0) = f(s_i) = 0, i = 1, 2, 3, 4 $, $ f(s) > 0 $ for $ s\in(s_1, s_2)\cup(0, s_3)\cup(s_4, +\infty), f(s) < 0 $ for $ s\in(-\infty, s_1)\cup(s_2, 0)\cup(s_3, s_4) $. Under some suitable assumptions on nonlinear terms, we prove the existence of unbounded continua of nodal solutions of this problem which bifurcate from the line of trivial solutions or from infinity, respectively.

Keywords

• nodal solutions
• bifurcation
• nonlocal problem
• eigenvalues