A complete classification of bifurcation diagrams for a class of (p,q)-Laplace equations

We study the bifurcation problem of positive solutions for the one-dimensional (p,q)-Laplace equation with nonlinear term u^r−1. There are five types of order relations for (p,q,r). We investigate the exact shape of the bifurcation curve in each type of the order relation. We prove that there are two types of bifurcation curves that are increasing, two types that are decreasing, and one that is not monotone and turns exactly once. Moreover, we study the asymptotic profile of the normalized solution u(x)/‖u‖_∞ as ‖u‖_∞→0 or ‖u‖_∞→∞ where ‖u‖_∞ denotes the L^∞-norm of u.