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.

Abstract. In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem:Here Ω is an annulus in R n , n ≥ 3, σ < 1 and q is a positive function in C γ loc (Ω), 0 < γ < 1, satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub-and supersolutions with Karamata regular variation theory.