  Two sharp existence and uniqueness theorems are presented for solutions of
multiple vortices arising in a six-dimensional brane-world supersymmetric gauge
field theory under the general gauge symmetry group $G=U(1)\times SU(N)$ and
with $N$ Higgs scalar fields in the fundamental representation of $G$.
Specifically, when the space of extra dimension is compact so that vortices are
hosted in a 2-torus of volume $|\Om|$, the existence of a unique multiple
vortex solution representing $n_1,...,n_N$ respectively prescribed vortices
arising in the $N$ species of the Higgs fields is established under the
explicitly stated necessary and sufficient condition \[ n_i<\frac{g^2v^2}{8\pi
N}|\Om|+\frac{1}{N}(1-\frac{1}{N}[\frac{g}{e}]^2)n,\quad i=1,...,N,] where $e$
and $g$ are the U(1) electromagnetic and SU(N) chromatic coupling constants,
$v$ measures the energy scale of broken symmetry, and $n=\sum_{i=1}^N n_i$ is
the total vortex number; when the space of extra dimension is the full plane,
the existence and uniqueness of an arbitrarily prescribed $n$-vortex solution
of finite energy is always ensured. These vortices are governed by a system of
nonlinear elliptic equations, which may be reformulated to allow a variational
structure. Proofs of existence are then developed using the methods of calculus
of variations.
