A new concept of reduced measure for nonlinear elliptic equations

Brezis, Haim;Marcus, Moshe;Ponce, Augusto
(2004) Comptes rendus - Mathématique — Vol. 339, n° 3, p. 169-174 (2004)

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Abstract
We study the existence of solutions of the nonlinear problem -Deltau+g(u) = mu in Omega, u = 0 on partial derivativeOhm, where mu is a Radon measure and g:R --> R is a nondecreasing continuous function with g(t) = 0, For Allt less than or equal to 0. Given g, Eq. (i) need not have a solution for every measure It, and we say that g is a good measure if (i) admits a solution. We show that for every mu there exists a largest good measure mu* less than or equal to mu. This reduced measure mu* has a number of remarkable properties. (C) 2004 Academie des sciences. Published by Elsevier SAS. All rights reserved.
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Brezis, H., Marcus, M., & Ponce, A. (2004). A new concept of reduced measure for nonlinear elliptic equations. Comptes rendus - Mathématique, 339(3), 169-174. https://doi.org/10.1016/j.crma.2004.05.012 (Original work published 2004)