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In this article, we prove existence results of positive solutions for thefollowing nonlinear elliptic problem with gradient terms: \begin{eqnarray*}\left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nablav) &{\rm in}\,\,\Omega,\\ (-\Delta)^\alpha v=g(x,u,v,\nabla u, \nabla v) &{\rmin}\,\,\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 someassumptions 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 Spruckand topological degree theory.

Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian