Unilateral Global Bifurcation for Fourth-Order Problems and Its Applications - Wenguo Shen (Discrete Dynamics in Nature and Society)
We will establish unilateral global bifurcation result for a class of fourth-order problems. Under some natural hypotheses on perturbation function, we show that (λk,0) is a bifurcation point of the above problems and there are two distinct unbounded continua, Ck+ and Ck-, consisting of the bifurcation branch Ck from (μk,0), where μk is the kth eigenvalue of the linear problem corresponding to the above problems. As the applications of the above result, we study the existence of nodal solutions for the following problems: x′′′′+kx′′+lx=rh(t)f(x),  0<t<1, x(0)=x(1)=x′(0)=x′(1)=0, where r∈R is a parameter and k,l are given constants; h(t)∈C([0,1],[0,∞)) with h(t)≢0 on any subinterval of [0,1]; and f:R→R is continuous with sf(s)>0 for s≠0. We give the intervals for the parameter r≠0 which ensure the existence of nodal solutions for the above fourth-order Dirichlet problems if f0∈[0,∞] or f∞∈[0,∞], where f0=lim|s|→0f(s)/s and f∞=lim|s|→+∞f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.