On the classification and evolution of bifurcation curves for a one-dimensional prescribed curvature problem with nonlinearity exp(aua+u)

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)′=λexp(aua+u),−L0  is a bifurcation parameter, and L,a>0  are two evolution parameters. We prove that, on (λ,‖u‖∞)-plane, for 036/17, 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.