\section{Introduction}
\setcounter{equation}{0}
In this paper we treat a 3-dimensional analogue of the vortex equations in
2 dimensions and look for solutions with spherical symmetry as described in
\S2. The domain considered is a 3-dimensional disk and we prescribe the
degree of the monopole at the boundary. Unlike in 2 dimensions, where
Ginzburg-Landau-type functionals appear either with or without gauge
potentials, the problem in 3 dimensions is well-posed only in gauge
theory. Without a curvature term in the action, a minimizing sequence
for the action would yield a trivial limit \cite{mar1}. In the
presence of a gauge potential, this problem is well-posed, natural and
has physical meaning.
The most general Yang-Mills-Higgs functional takes the form
\begin{equation}
\label{ymhgeneral}
\YMH(A,\phi) = \frac{\epsilon }{2}\| F\|^2_{L^2} +
\frac{\rho}{2}\| D_A\phi\|^2_{L^2} +
\frac{\lambda}{8} \|\,|\phi|^2-a^2\|^2_{L^2}\ ,
\end{equation}
for appropriate constants $\epsilon$, $\rho$, $\lambda$ and
$a$. Working on $\R^3$, one usually applies a rescaling of $\phi$, a
rescaling of space, and a rescaling of the action to set
$\epsilon=\rho=a=1$, so the action functional depends on a single
parameter, $\lambda$. On the unit ball, however, we cannot rescale
space, so we can only eliminate two of the four parameters. We
set $\rho=a=1$, and obtain a 2-paramater family of functionals
\begin{equation}
\label{ymh}
\YMH_{\epsilon,\lambda} (A,\,\phi)=
\frac{\epsilon }{2}\| F\|^2_{L^2(B^{3})} + \frac{1}{2}\| D_A\phi\|^2_{L^2(B^{3})} +
\frac{\lambda}{8} \|\,|\phi|^2-1\|^2_{L^2(B^{3})}\ .
\end{equation}
(Alternatively, we could work on a sphere of radius $R$. One can then
rescale to set $\epsilon=1$, at the cost of varying $R$. We then obtain
a 2-parameter family of functionals indexed by $\lambda$ and $R$.)
We know from the general theory for monopoles (cf. \cite {mar1} for
$\epsilon = 1,$ $\lambda \geq 0$) that there exists a minimum for this
functional which satisfies the Euler Lagrange equations
\begin{equation} \label{pde}
\begin{split}
\epsilon \, *D_A* F&=[D_A\phi,\,\phi]\\
*D_A*D_A\phi&=\frac{\lambda}{2}(|\phi|^2-1)\phi
\end{split}
\end{equation}
and suitable boundary conditions on $\partial B^{3}\equiv S^{2}$
(cf. Section 4 and \cite{mar1, mar2}), and is smooth.
In this paper we prove the existence, and describe the form, of
spherically symmetric solutions to these equations.
We note that, even for $\lambda=0$, these are not
solutions to the Bogomolnyi equations found in \cite{ja-ta}.
The Bogomolnyi solutions are obtained only in the limit
$\lambda \to 0$, $R \to \infty$, or equivalently
$\lambda \to 0$, $\epsilon \to 0$.