Unilateral Global Bifurcation for Fourth-Order Problems and Its Applications

Abstract

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+andCk-, consisting of the bifurcation branchCkfrom(μk,0), whereμkis thekth 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, wherer∈Ris a parameter andk,lare given constants;h(t)∈C([0,1],[0,∞))withh(t)≢0on any subinterval of[0,1]; andf:R→Ris continuous withsf(s)>0fors≠0.We give the intervals for the parameterr≠0which ensure the existence of nodal solutions for the above fourth-order Dirichlet problems iff0∈[0,∞]orf∞∈[0,∞],wheref0=lim|s|→0f(s)/sandf∞=lim|s|→+∞f(s)/s.We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.