Measure data problems for a class of elliptic equations with mixed absorption-reaction

We study the existence of nonnegative solutions to the Dirichlet problem $\CL^{_{^M}}_{p,q}u:=-Δu+u^p-M|\nabla u|^q=μ$ in a domain $Ω\subset\BBR^N$ where $μ$ is a nonnegative Radon measure, when $p>1$, $q>1$ and $M\geq 0$. We also give conditions under which nonnegative solutions of $\CL^{_{^M}}_{p,q}u=0$ in $Ω\setminus K$ where $K$ is a compact subset of $Ω$ can be extended as a solution of the same equation in $Ω$

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