Nodal solutions for a fourth-order two-point boundary value problem

Abstract

We consider boundary value problems of fourth-order differential equations of the form u ⁗ + β u ″ − α u = μ h ( x ) f ( u ) , 0 < x < r , u ( 0 ) = u ( r ) = u ″ ( 0 ) = u ″ ( r ) = 0 , where μ is a parameter, β ∈ ( − ∞ , ∞ ) , α ∈ [ 0 , ∞ ) are constants with r 2 β π 2 + r 4 α π 4 < 1 , h ∈ C ( [ 0 , r ] , [ 0 , ∞ ) ) with h ≢ 0 on any subinterval of [ 0 , r ] , f ∈ C ( R , R ) satisfies f ( u ) u > 0 for all u ≠ 0 , and lim u → − ∞ f ( u ) u = 0 , lim u → + ∞ f ( u ) u = f + ∞ , lim u → 0 f ( u ) u = f 0 for some f + ∞ , f 0 ∈ ( 0 , ∞ ) . We use bifurcation techniques to establish existence and multiplicity results of nodal solutions to the problem.