  We consider, in a smooth bounded multiply connected domain $\dom\subset\R^2$,
the Ginzburg-Landau energy $\d E_\v(u)=1/2\int_\dom{|\n
u|^2}+\frac{1}{4\v^2}\int_\dom{(1-|u|^2)^2}$ subject to prescribed degree
conditions on each component of $\p\dom$. In general, minimal energy maps do
not exist \cite{BeMi1}. When $\dom$ has a single hole, Berlyand and Rybalko
\cite{BeRy1} proved that for small $\v$ local minimizers do exist. We extend
the result in \cite{BeRy1}: $\d E_\v(u)$ has, in domains $\dom$ with $2,3,...$
holes and for small $\v$, local minimizers. Our approach is very similar to the
one in \cite{BeRy1}; the main difference stems in the construction of test
functions with energy control.

Analysis of PDEs