Existence of boundary blow-up solutions for a class of quasilinear elliptic systems with critical case

Abstract

We consider the quasilinear elliptic system div ( | ∇ u | p - 2 ∇ u ) = u m 1 v n 1 , div ( | ∇ v | q - 2 ∇ v ) = u m 2 v n 2 in Ω , where m 1 > p - 1 , n 2 > q - 1 , m 2 , n 1 > 0 , and Ω ⊂ R N is a smooth bounded domain, subject to three different types of Dirichlet boundary conditions: u = λ , v = μ or u = v = + ∞ or u = + ∞ , v = μ on ∂ Ω , where λ , μ > 0 . Under several hypotheses on the parameters m 1 , n 1 , m 2 , n 2 which is a critical case, we show that the existence of positive solutions.