Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations

We study the positive solution ({u(r,rho)}) of the quasilinear elliptic equation

$$begin{cases}r^{-(gamma-1)}(r^{alpha}|u^{prime}|^{beta-1}u^{prime})^{prime}+|u|^{p-1}u=0, & 0 < r < infty, u(0) = rho > 0, u^{prime}(0)=0.end{cases}$$

This class of differential operators includes the usual Laplace, m-Laplace, and k-Hessian operators in the space of radial functions. The equation has a singular positive solution u ^*(r) under certain conditions on ({alpha}), ({beta}), ({gamma}), and p. A generalized Joseph–Lundgren exponent, which we denote by ({p^*_{JL}}), is obtained. We study the intersection numbers between ({u(r,rho)}) and u ^*(r) and between ({u(r,rho_0)}) and ({u(r,rho_1)}), and see that ({p^*_{JL}}) plays an important role. We also determine the bifurcation diagram of the problem

$$begin{cases}r^{-(gamma-1)}(r^{alpha}|u^{prime}|^{beta-1}u^{prime})^{prime} + lambda(u+1)^p=0, & 0 < r < 1, u(r) > 0, & 0 le r < 1, u^{prime}(0)=0, u(1)=0.end{cases}$$

The main technique used in the proofs is a phase plane analysis.