Large solutions for an elliptic system of quasilinear equations

In this paper we consider the quasilinear elliptic system Δpu=uavb, Δpv=ucve in a smooth bounded domain Ω⊂RN, with the boundary conditions u=v=+∞ on ∂Ω. The operator Δp stands for the p-Laplacian defined by Δpu=div(|∇u|p−2∇u), p>1, and the exponents verify a,e>p−1, b,c>0 and (a−p+1)(e−p+1)⩾bc. We analyze positive solutions in both components, providing necessary and sufficient conditions for existence. We also prove uniqueness of positive solutions in the case (a−p+1)(e−p+1)>bc and obtain the exact blow-up rate near the boundary of the solution. In the case (a−p+1)(e−p+1)=bc, infinitely many positive solutions are constructed.