On the classification and evolution of bifurcation curves for a one-dimensional prescribed curvature problem with nonlinearity exp(aua+u). (November 2016) - Cheng, Yan-Hsiou; Hung, Kuo-Chih; Wang, Shin-Hwa
Abstract: We study the classification and evolution of bifurcation curves of positive solutions u for the one-dimensional prescribed curvature problem − ( u ′ ( x ) 1 + ( u ′ ( x ) ) 2 ) ′ = λ exp ( a u a + u ), − L < x < L, u ( − L ) = u ( L ) = 0, where λ > 0 is a bifurcation parameter, and L, a > 0 are two evolution parameters. We prove that, on ( λ, ‖ u ‖ ∞ ) -plane, for 0 < a ≤ 36 / 17 ≈ 2.118, the bifurcation curve is ⊃ -shaped. While for a > 36 / 17, the bifurcation curve is ⊃ -shaped or reversed ε -like shaped. In particular, for a > a ∗ ∗ ≈ 4.107, the bifurcation curve is (i) ⊃ -shaped if L > 0 small enough and (ii) reversed ε -like shaped if L is large enough.,