A note on entire large solutions to semilinear elliptic systems of competitive type

Abstract

We consider the elliptic system $\Delta u=p\left(\left|x\right|\right)u^a{v}^b$, $\Delta v=q\left(\left|x\right|\right)u^c{v}^d$ on $R^n$ ($n\ge 3$) where $a$, $b$, $c$, $d$ are positive constants with $min\{a,d\}>1$, and the functions $p$ and $q$ are positive and continuous. We establish conditions on $p$ and $q$, along with the exponents $a,b,c,d$, which ensure the existence of a positive entire solution satisfying ${lim}_{\left|x\right|\to \infty }u\left(x\right)={lim}_{\left|x\right|\to \infty }v\left(x\right)=\infty$.