  In this article, we prove existence results of positive solutions for the
following nonlinear elliptic problem with gradient terms: \begin{eqnarray*}
\left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nabla
v) &{\rm in}\,\,\Omega,\\ (-\Delta)^\alpha v=g(x,u,v,\nabla u, \nabla v) &{\rm
in}\,\,\Omega,\\ u=v=0\,\,&{\rm in}\,\,\R^N\setminus\Omega,
  \end{array}
  \right.
  \end{eqnarray*} where $(-\Delta)^\alpha$ denotes the fractional Laplacian and
$ \Omega $ is a smooth bounded domain in $ \R^N $. It shown that under some
assumptions on $ f $ and $ g $, the problem has at least one positive solution
$(u,v)$. Our proof is based on the classical scaling method of Gidas and Spruck
and topological degree theory.
