Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒp,qM⁢u:=-Δ⁢u+up-M⁢|∇⁡u|q=μ{{\mathcal{L}}^{{M}}_{p,q}u:=-\Delta u+u^{p}-M|\nabla u|^{q}=\mu} in a domain Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} where μ is a nonnegative Radon measure, when p>1{p>1}, q>1{q>1} and M≥0{M\geq 0}. We also give conditions under which nonnegative solutions of ℒp,qM⁢u=0{{\mathcal{L}}^{{M}}_{p,q}u=0} in Ω∖K{\Omega\setminus K}, where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.