Elliptic equations with nonlinear absorption depending on the solution and its gradient

We study positive solutions of equation (E1) − Δ u + u p| ∇ u | q = 0 ( 0 ⩽ p , 0 ⩽ q ⩽ 2 , p + q > 1 ) and (E2) − Δ u + u p + | ∇ u | q = 0 ( p > 1 , 1 < q ⩽ 2 ) in a smooth bounded domain Ω ⊂ R N . We obtain a sharp condition on p and q under which, for every positive, finite Borel measure μ on ∂ Ω , there exists a solution such that u = μ on ∂ Ω . Furthermore, if the condition mentioned above fails, then any isolated point singularity on ∂ Ω is removable, namely, there is no positive solution that vanishes on ∂ Ω everywhere except at one point. With respect to (E2), we also prove uniqueness and discuss solutions that blow up on a compact subset of ∂ Ω . In both cases, we obtain a classification of positive solutions with an isolated boundary singularity. Finally, in Appendix A a uniqueness result for a class of quasilinear equations is provided. This class includes (E1) when p = 0 but not the general case.