  We prove the existence of the first eigenvalue and an associated
eigenfunction with Dirichlet condition for the complex Monge-Amp\`ere operator
on a bounded strongly pseudoconvex domain in $\C^n$. We show that the
eigenfunction is plurisubharmonic, smooth with bounded Laplacian in $\Omega$
and boundary values $0$. Moreover it is unique up to a positive multiplicative
constant.
  To this end, we follow the strategy used by P.L. Lions in the real case.
However, we have to prove a new theorem on the existence of solutions for some
special complex degenerate Monge-Amp\`ere equations. This requires establishing
new a priori estimates of the gradient and Laplacian of such solutions using
methods and results of L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck
\cite{CKNS85} and B. Guan \cite{GuanB98}.
  Finally we provide a Pluripotential variational approach to the problem and
using our new existence theorem, we prove a Rayleigh quotient type formula for
the first eigenvalue of the complex Monge-Amp\`ere operator.
