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### Boundary Value Problem for $r^2 d^2 f/dr^2 + f = f^3$ (III): Global Solution and Asymptotics

Based on the results in the previous papers that the boundary value problem
$y'' - y' + y = y^3, y(0) = 0, y(\infty) =1$ with the condition $y(x) > 0$ for
$0<x<\infty$ has a unique solution $y^*(x)$, and $a^*= y^{*^{'}}(0)$ satisfies
$0<a^*<1/4$, in this paper we show that $y'' - y' + y = y^3, -\infty < x < 0$,
with the initial conditions $y(0) = 0, y'(0) = a^*$ has a unique solution by
using functional analysis method. So we get a globally well defined bounded
function $y^*(x), -\infty < x < +\infty$. The asymptotics of $y^*(x)$ as $x \to
- \infty$ and as $x \to +\infty$ are obtained, and the connection formulas for
the parameters in the asymptotics and the numerical simulations are also given.
Then by the properties of $y^*(x)$, the solution to the boundary value problem
$r^2 f'' + f = f^3, f(0)= 0, f(\infty)=1$ is well described by the asymptotics
and the connection formulas.Comment: 11 pages, 2 fingure

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

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.Comment: 12 page

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