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

Abstract: 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.