| Popis: |
We study the bifurcation curves of positive solutions of the boundary value problem { u ″ ( x ) + f e ( u ( x ) ) = 0 , − 1 x 1 , u ( − 1 ) = u ( 1 ) = 0 , where f e ( u ) = g ( u ) − e h ( u ) , e ∈ 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 ( e , ‖ u ‖ ∞ ) -plane; that is, the bifurcation curve has exactly two turning points at some points ( e , ‖ u e ‖ ∞ ) and ( e ∗ , ‖ u e ∗ ‖ ∞ ) such that e e ∗ and ‖ u e ‖ ∞ ‖ u e ∗ ‖ ∞ . In addition, we prove that e ∗ > 0 . Thus the exact number of positive solutions can be precisely determined by the values of e and e ∗ . We give an application to the two-parameter bifurcation problem { u ″ ( x ) + λ ( 1 + u 2 − e u 3 ) = 0 , − 1 x 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ , e are two positive bifurcation parameters. Some new results are obtained. |
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