Existence and asymptotic behavior of ground state solutions of semilinear elliptic system

In this article, we take up the existence and the asymptotic behavior of entire bounded positive solutions to the following semilinear elliptic system:-Δu = a_{1}(x)u^{\alpha}v^{r}, x\in\mathbb{R}^{n} (n\geq 3), -Δv = a_{2}(x)v^{\beta}u^{s}, x\in\mathbb{R}^{n}, u,v ¿ 0 in \mathbb{R}^{n}, \lim_{|x|\rightarrow+\infty}u(x) = \lim_{|x|\rightarrow+\infty}v(x)=0,where {\alpha,\beta<1}, {r,s\in\mathbb{R}} such that {\nu:=(1-\alpha)(1-\beta)-rs>0}, and the functions a_{1}, a_{2} are nonnegative in {\mathcal{C}^{\gamma}_{\mathrm{loc}}(\mathbb{R}^{n})} (0¡γ¡1) and satisfy some appropriate assumptions related to Karamata regular variation theory.