Abstract
We study the classification and evolution of bifurcation curves of positive solutions u for the one-dimensional prescribed curvature problem{−([Formula presented]) <sup>′</sup> =λexp([Formula presented]), −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‖ <sub>∞</sub> )-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 <sup>∗∗</sup> ≈4.107, the bifurcation curve is (i) ⊃-shaped if L>0 small enough and (ii) reversed ε-like shaped if L is large enough.