Large solutions for an elliptic system of quasilinear equations

Abstract

In this paper we consider the quasilinear elliptic system Δ p u = u a v b , Δ p v = u c v e in a smooth bounded domain Ω ⊂ R N , with the boundary conditions u = v = + ∞ on ∂ Ω. The operator Δ p stands for the p-Laplacian defined by Δ p u = 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 ) ⩾ b c . 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 ) > b c and obtain the exact blow-up rate near the boundary of the solution. In the case ( a − p + 1 ) ( e − p + 1 ) = b c , infinitely many positive solutions are constructed.