Nodal solutions of second-order boundary value problems with superlinear or sublinear nonlinearities

We consider boundary value problems for nonlinear second-order differential equations of the form u″+a(t)f(u)=0,t∈(0,1),u(0)=u(1)=0,u(0)=u(1)=0, where a∈C1([0,1],[0,∞))a∈C1([0,1],[0,∞)) with a(t0)>0a(t0)>0 for some t0∈[0,1]t0∈[0,1] and f:R→Rf:R→R is continuous with f(s)s>0f(s)s>0 for s≠0s≠0. We establish existence and multiplicity results for nodal solutions of the problems if either f0=0f0=0, f∞=∞f∞=∞ or f0=∞f0=∞, f∞=0f∞=0, where f(s)/sf(s)/s approaches f0f0 and f∞f∞ as ss approaches 0 and ∞∞, respectively. We use bifurcation techniques to prove our main results.