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

We study the classification and evolution of bifurcation curves of positive solutions u for the one-dimensional prescribed curvature problem [GRAPHICS] where lambda > 0 is a bifurcation parameter, and L, a > 0 are two evolution parameters. We prove that, on (lambda, parallel to u parallel to(infinity))-plane, for 0 < a <= 36/17 approximate to 2.118, the bifurcation curve is superset of-shaped. While for a > 36/17, the bifurcation curve is superset of-shaped or reversed epsilon-like shaped. In particular, for a > a** approximate to 4.107, the bifurcation curve is (i) superset of-shaped if L > 0 small enough and (ii) reversed epsilon-like shaped if L is large enough. (C) 2016 Elsevier Ltd. All rights reserved.