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

Abstract 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 , where a ∈ C 1 ( [ 0 , 1 ] , [ 0 , ∞ ) ) with a ( t 0 ) > 0 for some t 0 ∈ [ 0 , 1 ] and f : R → R is continuous with f ( s ) s > 0 for s ≠ 0 . We establish existence and multiplicity results for nodal solutions of the problems if either f 0 = 0 , f ∞ = ∞ or f 0 = ∞ , f ∞ = 0 , where f ( s ) / s approaches f 0 and f ∞ as s approaches 0 and ∞ , respectively. We use bifurcation techniques to prove our main results.