Boundary Singular Problems for Quasilinear Equations Involving Mixed Reaction–Diffusion.

We study the existence of solutions to the problem - Δ u + u p - M ∇ u q = 0 in Ω , (1) u = μ on ∂ Ω in a bounded domain Ω, where p > 1, 1 < q < 2, M > 0, μ is a nonnegative Radon measure in ∂Ω, and the associated problem with a boundary isolated singularity at a ∈ ∂Ω, - Δ u + u p - M ∇ u q = 0 in Ω , (2) u = 0 on ∂ Ω \ α. The difficulty lies in the opposition between the two nonlinear terms which are not on the same nature. Existence of solutions to (1) is obtained under a capacitary condition μ K ≤ c min cap 2 p , p ′ ∂ Ω , cap 2 - q q , q ′ ∂ Ω for\;all\;compacts\; K ⊂ ∂ Ω. Problem (2) depends on several critical exponents on p and q as well as the position of q with respect to 2 p p + 1 . [ABSTRACT FROM AUTHOR]