Nonlinear elliptic equations with measures revisited [article]

Haïm Brezis, Moshe Marcus, Augusto C. Ponce
2013 arXiv   pre-print
We study the existence of solutions of the nonlinear problem { -Δ u + g(u) & = μ & & in Ω, u & = 0 & & on ∂Ω, . where μ is a Radon measure and g : R→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^* ≤μ. This reduced measure has a number of remarkable properties.
arXiv:1312.6495v1 fatcat:bmggjjxp6jefbibf45vxghidji