A theorem on reversed S-shaped bifurcation curves for a class of boundary value problems and its application

Abstract

We study the bifurcation curves of positive solutions of the boundary value problem { u ″ ( x ) + f ε ( u ( x ) ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where f ε ( u ) = g ( u ) − ε h ( u ) , ε ∈ R is a bifurcation parameter, and functions g , h ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) satisfy five hypotheses presented herein. Assuming these hypotheses on fixed g and h , we prove that the bifurcation curve is reverse S-shaped on the ( ε , ‖ u ‖ ∞ ) -plane; that is, the bifurcation curve has exactly two turning points at some points ( ε ̃ , ‖ u ε ̃ ‖ ∞ ) and ( ε ∗ , ‖ u ε ∗ ‖ ∞ ) such that ε ̃ < ε ∗ and ‖ u ε ̃ ‖ ∞ < ‖ u ε ∗ ‖ ∞ . In addition, we prove that ε ∗ > 0 . Thus the exact number of positive solutions can be precisely determined by the values of ε ̃ and ε ∗ . We give an application to the two-parameter bifurcation problem { u ″ ( x ) + λ ( 1 + u 2 − ε u 3 ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ , ε are two positive bifurcation parameters. Some new results are obtained.