Large solutions to an anisotropic quasilinear elliptic problem

Abstract

In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem: divx(|∇xu| p−2∇xu)(x, y) + divy(|∇yu| q−2∇yu)(x, y) = u r (x, y) in a bounded domain Ω⊂RN×RMΩ⊂RN×RM together with the boundary condition u (x, y) = ∞ on ∂Ω. We prove that the necessary and sufficient condition for the existence of a solution u∈W1,p,qloc(Ω)u∈Wloc1,p,q(Ω) to this problem is r > max{p−1, q−1}. Assuming that r > q−1 ≥ p−1 > 0 we will show that the exponent q controls the blow-up rates near the boundary in the sense that all points of ∂Ω share the same profile, that depends on q and r but not on p, with the sole exception of the vertical points (where the exponent p plays a role).