The shooting method and the analysis of the target map via the degree theory

The shooting method and the analysis of the target map via the degree theory

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Congming Li, University of Colorado at Boulder
Fine Hall 314

We introduce and analysis the ‘target map’ for the shooting method. For a large class of elliptic systems as well as more general dynamic systems, we show that the target map is onto via the degree theory. The target map is onto implies that we can shoot to any desired position.  Applying our result to a motivating example, we obtain the existence of global positive solutions to the Hardy-Littlewood-Sobolev (also known as Lane-Emden) type system:$$\begin{cases}& (-\Delta)^k u = v^p, \text{in $\mathbb{R}^n$},\\& (-\Delta)^k u = v^q, \text{in $\mathbb{R}^n$},\\\end{cases}$$in the critical and supercritical case $\frac{1}{p+1} + \frac{1}{q+1}\ge\frac{n-2k}{n+2k}$. On the other hand, the Lane-Emden conjecture state that when $k = 1$, the above system admits no positive solution in the subcritical case $\frac{1}{p+1}+\frac{1}{q+1} < \frac{n−2k}{n+2k}$. The Lane-emden conjecture has been proved for many cases and no counter-example has been found yet.This new approach to the shooting method is simple and powerful.  We expect many more applications of it.