Existence of entire positive solutions for semilinear elliptic systems with gradient term

Abstract

Under the Keller–Osserman condition on \({\Sigma_{j=1}^{2}f_{j}}\), we show the existence of entire positive solutions for the semilinear elliptic system \({\Delta u_{1}+|\nabla u_{1}|=p_{1}(x)f_{1}(u_{1},u_{2}), \Delta u_{2}+|\nabla u_{2}|=p_{2}(x)f_{2}(u_{1},u_{2}),x \in \mathbb{R}^{N}}\), where \({p_{j}(j=1, 2):\mathbb{R}^{N} \rightarrow [0,\infty)}\) are continuous functions.