Singular quasilinear and Hessian equations and inequalities

We solve the existence problem in the renormalized, or viscosity sense, and obtain global pointwise estimates of solutions for quasilinear and Hessian equations with measure coefficients and data, including the following model problems:−Δpu=σuq+μ,Fk[−u]=σuq+μ,u⩾0, on RnRn, or on a bounded domain Ω⊂RnΩ⊂Rn. Here ΔpΔp is the p  -Laplacian defined by Δpu=div(∇u|∇u|p−2)Δpu=div(∇u|∇u|p−2), and Fk[u]Fk[u] is the k  -Hessian, i.e., the sum of the k×kk×k principal minors of the Hessian matrix D2uD2u (k=1,2,…,nk=1,2,…,n); σ and μ are general nonnegative measurable functions (or measures) on Ω.