Multiplicity results for second-order two-point 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 ( [ 0 , 1 ] , ( 0 , ∞ ) ) and f : R → R is continuous and satisfies f ( s ) s > 0 for s ≠ 0 . We establish existence and multiplicity results for nodal solutions to 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.