Nonlinear elliptic equations with measures revisited

We study the existence of solutions of the nonlinear problem $\left\{ \begin{alignedat}{2} -Δu + g(u) & = μ& & \quad \text{in } Ω,\\ u & = 0 & & \quad \text{on } \partial Ω, \end{alignedat} \right.$ where $μ$ is a Radon measure and $g : \mathbb{R} \to \mathbb{R}$ is a nondecreasing continuous function with $g(0) = 0$. This equation need not have a solution for every measure $μ$, and we say that $μ$ is a good measure if the Dirichlet problem above admits a solution. We show that for every $μ$ there exists a largest good measure $μ^* \leq μ$. This reduced measure has a number of remarkable properties.

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