Unilateral Global Bifurcation for Fourth-Order Problems and Its Applications

We will establish unilateral global bifurcation result for a class of fourth-order problems. Under some natural hypotheses on perturbation function, we show that (lambda(k), 0) is a bifurcation point of the above problems and there are two distinct unbounded continua, C-k(+) and C-k(-), consisting of the bifurcation branch C-k from (mu(k), 0), where Ilk 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 epsilon R is a parameter and k,I are given constants; h(t) epsilon C([0,1], [0, con 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 t 0 which ensure the existence of nodal solutions for the above fourth-order Dirichlet problems if f(0) epsilon [0,infinity] or f(infinity), epsilon [0, infinity], where f(infinity) = lim(vertical bar s vertical bar -> 0)f(s)/s and lim(vertical bar s vertical bar ->infinity)f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.