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Ponce, Ha\\\"im Brezis, Moshe Marcus","submitted_at":"2013-12-23T09:40:28Z","abstract_excerpt":"We study the existence of solutions of the nonlinear problem $$ \\left\\{ \\begin{alignedat}{2} -\\Delta u + g(u) & = \\mu & & \\quad \\text{in } \\Omega,\\\\ u & = 0 & & \\quad \\text{on } \\partial \\Omega, \\end{alignedat} \\right. $$ where $\\mu$ 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 $\\mu$, and we say that $\\mu$ is a good measure if the Dirichlet problem above admits a solution. We show that for every $\\mu$ there exists a largest good measure $\\mu^* \\leq \\mu$. 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