\section{Existence of spherically symmetric monopoles}
\setcounter{equation}{0}
Our basic existence result is
\begin{theorem}
\label{existence}
For all values of $\lambda\geq 0,$ $\epsilon > 0,$
there exists a symmetric solution of
\begin{equation} \label{system}
\left\{ \begin{split}
&\epsilon\,*D_A* F=[D_A\phi,\,\phi] \qquad\text { on } B^{3}\\
&*D_A*D_A\phi=\frac{\lambda}{2}(|\phi|^2-1)\phi \qquad\text { on } B^{3}\\
&(D\phi)_{\tau} = 0 \qquad\text { on } \partial B^{3} \\
&\vert \varphi\vert = 1 \qquad\text { on } \partial B^{3}
\end{split}\right.
\end{equation}
\end{theorem}
{\bf Observation:}
These equations do not reduce to the Bogomolnyi equations, even if
$\lambda= 0.$ The Bogolmolnyi argument involves an integration by parts;
on a finite domain, this results in a boundary contribution \cite{ja-ta}.
\begin{proof} Because of the derivative terms in the action, the natural
space for $\gamma$ (denoted ${\mathcal H}_\gamma$) is $H^1(0,1)$, while the
natural space ${\mathcal H}_\varphi$ for $\varphi$ is
the weighted Sobolev space $H^1((0,1),r^2 dr)$.
By the Sobolev embedding theorem, functions in ${\mathcal H}_\gamma$
are continuous on $[0,1]$. Functions in ${\mathcal H}_\varphi$ are
continuous on $(0,1]$, but may not have a limit at $r=0$.
We may therefore apply boundary conditions to
$\gamma$ at $r=0$ and at $r=1$, and to $\varphi$ at $r=1$.
Let
\begin{equation}
{\mathcal F}=\{(\gamma, \varphi)\in {\mathcal H}_\gamma \times {
\mathcal H}_\varphi \;:\;
\gamma(1)=-\frac{1}{2},\,\gamma(0)=0,
\, \varphi (1) = 1. \}
\end{equation}
%
%and let
%
%\begin{equation}
% \label{action}
% \begin{split}
%{\mathcal S}(\gamma, \varphi)
%= & \int_0^1 [\ 2\epsilon \left({\gamma^{\prime}}^{2}
%+
%\frac {2}{r^2} (\gamma^{2} +\gamma)^{2}\right) +
%r^2 {\varphi^{\prime}}^{2} +
%2 \varphi^2 (1+2\gamma)^{2} +\lambda
%r^{2}(\varphi^{2}-1)^{2}\ ] dr \\
% = & \SYMH (\gamma, \varphi)/4\pi.
%\end{split}
%\end{equation}
The action functional \eqref{symhgamma} is well-defined on ${\mathcal
F}$, and is finite whenever $\varphi$ is bounded. In particular, $\mu
\equiv Inf_{\mathcal F}\,{\mathcal S}$ is finite. We follow the
direct method in the calculus of variations. That is, take a
minimizing sequence for ${\mathcal S}$, show that it converges weakly
in ${\mathcal F}$, and then show that the weak limit minimizes the
action and so solves the Euler-Lagrange equations.
Let $(\gamma_n,\varphi_n)$ be a minimizing sequence for ${\mathcal
S}$. Since $\lambda \ge 0$, the action is not increased if we
make the replacement
%
\begin{equation}
\varphi(r) \to
\begin{cases}
-1, & \hbox{if }\varphi(r) < -1; \cr
\varphi(r), & \hbox{if }-1 \le \varphi(r) \le 1; \cr
1, & \hbox{if }\varphi(r) > 1.
\end{cases}
\end{equation}
%
As a result, we can assume that each $\varphi_n(r)$ is bounded in
magnitude by 1. Under these circumstances, the sequence $(\gamma_n,
\varphi_n)$ is bounded in ${\mathcal F} \subset {\mathcal H}_\gamma
\times {\mathcal H}_\varphi$. However, balls in ${\mathcal H}_\gamma$
are weakly compact, as are balls in ${\mathcal H}_\varphi$, so the
pair $(\gamma_n, \varphi_n)$ converges weakly in ${\mathcal H}_\gamma
\times {\mathcal H}_\varphi$ to a limit $(\gamma_\infty,
\varphi_\infty)$.
By Sobolev, $\gamma_n(r)$ and $\varphi_n(r)$ converge pointwise to
$\gamma_\infty(r)$ and $\varphi_\infty(r)$, so the limiting values
$\gamma(0)$, $\gamma(1)$, and $\varphi(1)$ are preserved, and
$(\gamma_\infty, \varphi_\infty) \in {\mathcal F}$. Moreover, terms in
${\mathcal S}(\gamma_n,\varphi_n)$ that don't involve derivatives
converge to the corresponding terms in ${\mathcal S}
(\gamma_\infty,\varphi_\infty)$. The derivative terms are
quadratic, hence weakly semicontinuous. As a result, ${\mathcal
S}(\gamma_\infty, \varphi_\infty)$ is bounded above by $\mu$, and
therefore must equal $\mu$.
Showing that $\gamma_\infty$ and $\varphi_\infty$ satisfy the
Euler-Lagrange equations \eqref{ode} is then a standard exercise in
the calculus of variations. Smoothness of $(\gamma_\infty,
\varphi_\infty)$ away from $r=0$ follows by elliptic regularity of the
equations \eqref{ode}. Smoothness at $r=0$ follows from regular
singular-point analysis, combined with the fact that both functions
are bounded (see \S 6 for details).
This in turn implies that the
connection and Higgs field $(A,\phi)$ constructed from $(\gamma_\infty,
\varphi_\infty)$ comprise a smooth, symmetric classical solution to
the PDE system \eqref{system}. (Alternatively, one can establish
regularity of $(A,\phi)$ from the ellipticity of the PDE system
\eqref{system}, since we are
working in a gauge with $d^* A=0$.)
\end{proof}