In this paper, some suffcient conditions under which the quasilinear elliptic system −div(|∇u|p−2∇u)=um1vn1;−div(|∇u|q−2∇u) =um2vn2 in RN (N¿3) has no radially symmetric positive solution is derived. Then by using this non-existence result, blow-up estimates for a class of quasilinear reaction-diffusion systems ut=div(|∇u|p−2∇u)+um1vn1; vt=div(|∇v|q−2∇v)+um2vn2 with the homogeneous Dirichlet boundary value conditions are obtained. Copyright ? 2003 John Wiley & Sons, Ltd.