Asymptotic behaviour of elliptic systems with mixed absorption and source terms

Abstract

We study the limit behaviour near the origin of the nonnegative solutions of the semilinear elliptic system $ \cases{ -\Delta u+\vert x\vert ^{a}v^{p}=0,\cr \Delta v+\vert x\vert ^{b}u^{q}=0,\cr}\quad \hbox{in}\ \mathbb{R}^{N}\ (N\geqslant 3), $ where $p,q,a,b\in \mathbb{R},$ with $p,q>0$ , $pq\neq 1$ . We give a priori estimates without any restriction on the values of $p$ and $q$ .