Global bifurcation and nodal solutions for a Sturm-Liouville problem with a nonsmooth nonlinearity

In this paper, we shall establish the unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity of a class of nonlinear Sturm-Liouville problems with nondifferentiable nonlinearity, respectively. As applications of the above results, we shall determine the interval of r, in which there exist nodal solutions for the following problem{u″(t)+ra(t)f(u)=0,t∈(0,1),u(0)=u(1)=0, where a(t) is a positive continuous function on [0,1], f∈C(R,R) but is not necessarily differentiable at the origin and infinity. Moreover, as a special case of the above problem, we shall prove more details about the existence of nodal solutions for a class of half-linear eigenvalue problems.