A degree theory approach for the shooting method
arXiv:1301.6232
Abstract
The classical shooting-method is about finding a suitable initial shooting positions to shoot to the desired target. The new approach formulated here, with the introduction and the analysis of the `target map' as its core, naturally connects the classical shooting-method to the simple and beautiful topological degree theory. We apply the new approach, to a motivating example, to derive the existence of global positive solutions of the Hardy-Littlewood-Sobolev (also known as Lane-Emden) type system: [{{aligned} &(-\triangle)^ku(x) = v^p(x), \,\, u(x)>0 \quad\text{in}\quad\mathbb{R}^n, & (-\triangle)^k v(x) =u^q(x), \,\, v(x)>0 \quad\text{in}\quad\mathbb{R}^n, p, q>0, {aligned}.] in the critical and supercritical cases . Here we derive the existence with the computation of the topological degree of a suitably defined target map. This and some other results presented in this article completely solved several long-standing open problems about the existence or non-existence of positive entire solutions.