Boundary Value Problem for $r^2 {d^2 f/dr^2} + f = f^3$ (I): Existence and Uniqueness

TL;DR

This paper proves the existence and uniqueness of solutions to a nonlinear boundary value problem involving a differential equation, using topological and variational methods, with implications for Yang-Mills equations.

Contribution

The paper introduces a topological shooting method and variation technique to establish existence and uniqueness for a specific nonlinear boundary value problem.

Findings

Existence of solutions proved via topological shooting argument.

Uniqueness established through variation method.

Foundation for analyzing global solutions and asymptotics in subsequent work.

Abstract

In this paper we study the equation $r^2 {d^2 f/dr^2} + f = f^3$ with the boundary conditions $f(1)=0$, $f(\infty)=1$ and $f(r) > 0$ for $1<r<\infty$. The existence of the solution is proved by using topological shooting argument. And the uniqueness is proved by variation method. Using the asymptotics of $f(r)$ as $r \to 1$, in the following papers we will discuss the global solution for $0<r<\infty$, and give explicit asymptotics of $f(r)$ as $r \to 0$ and as $r \to \infty$, and the connection formulas for the parameters in the asymptotics. Based on these results, we will solve the boundary value problem $f(0) =0$, $f(\infty) =1$, which is the goal of this work. Once people discuss the regular solution of this equation, this boundary value problem must be considered. This problem is useful to study the Yang-Mills potential related equations, and the method used for this equation is applicible to other similar equations.