Global bifurcation result and nodal solutions for Kirchhoff-type equation

We investigate the global structure of nodal solutions for the Kirchhoff-type problem (a + b 0 |u'|2dx)u'' = lambda f (u), x E (0,1), u(0) = u(1) = 0 , where a > 0, b > 0 are real constants, lambda is a real parameter. f E C(R, R) and there exist four constants s(1) < s(2) < 0 < s(3) < s(4) such that f (0) = f(si) = 0, i = 1, 2, 3, 4, f(s) > 0 for s E (s(1), s(2)) U (0, s3) U (s4, +co), f(s) < 0 for s E (-infinity, s(1)) U (s(2), 0) U (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.