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Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian (1511.03066v1)

Published 10 Nov 2015 in math.AP

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}\,\,\RN\setminus\Omega, \end{array} \right. \end{eqnarray*} where $(-\Delta)\alpha$ denotes the fractional Laplacian and $ \Omega $ is a smooth bounded domain in $ \RN $. 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.

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