Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications - Wenguo Shen; Tao He (Discrete Dynamics in Nature and Society)
We establish a unilateral global bifurcation result from interval for a class of fourth-order problems with nondifferentiable nonlinearity. By applying the above result, we firstly establish the spectrum for a class of half-linear fourth-order eigenvalue problems. Moreover, we also investigate the existence of nodal solutions for the following half-linear fourth-order problems: x″″=αx++βx-+ratfx, 0<t<1, x(0)=x(1)=x″(0)=x″(1)=0, where r≠0 is a parameter, a∈C([0,1],(0,∞)), x+=max⁡{x,0}, x-=-min⁡{x,0}, α,β∈C[0,1], and f∈C(R,R), sf(s)>0, for s≠0. We give the intervals for the parameter r which ensure the existence of nodal solutions for the above fourth-order half-linear problems if f0∈[0,∞) or f∞∈[0,∞], where f0=lims→0f(s)/s and f∞=lims→+∞f(s)/s. We use the unilateral global bifurcation techniques and the approximation of connected components to prove our main results.