On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type

We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation {-(u'/root 1 + u'(2))' = lambda(u/1 + u)(p), -L < x < L, u(-L) = u(L) = 0, where. is a bifurcation parameter, and L, p > 0 are two evolution parameters. We prove that on the (lambda, parallel to u parallel to(infinity))-plane, for 0 < p <= root 2/4, the bifurcation curve is superset of-shaped bifurcation starting from (0, 0). And for p = 1, f(u) = u/1+u is a logistic function, then the bifurcation curve is also superset of-shaped bifurcation starting from (pi(2)/4L(2), 0). While for p > 1, the bifurcation curve is reversed e-like shaped bifurcation if L > L*, and isexactly decreasing for lambda > lambda* if 0 < L < L-*.