@inbook{10.2307/j.ctt7s1f9.7,
ISBN = {9780691129556},
URL = {http://www.jstor.org/stable/j.ctt7s1f9.7},
abstract = {Let Ω ⊂ ℝNbe a bounded domain with smooth boundary. Letg: ℝ → ℝ be a continuous, nondecreasing function such thatg(0) = 0. In this paper we are concerned with the problem\[\left\{ \begin{array}{rl} - \Delta u + g(u) = \mu & \text{in}\ \Omega,\\ u = 0 & \text{on}\ \partial \Omega, \\ \end{array} \right. \caption{(4.0.1)}\]where μ is a measure. The study of (4.0.1) when μ ∈L¹(Ω) was initiated by Brezis-Strauss [BS]; their main result asserts that foreveryμ ∈L¹ andevery gas above, problem (4.0.1) admits a unique weak solution (see Theorem 4.B.2 in Appendix 4B below). The right concept of weak solution is the following:\[\left\{ \begin{array}{l} u \in L^1 (\Omega), g(u) \in L^1 (\Omega )\ \text{and}\\ - \int_{\Omega} u\Delta \zeta + \int_{\Omega } g(u)\zeta = \int_{\Omega} \zeta\, d\mu \quad \forall \zeta \in C^2 (\bar{\Omega}), \zeta = 0\ \text{on}\ \partial \Omega. \\ \end{array} \right. \caption{(4.0.2)}\]It will be convenient to write\[C_0(\bar{\Omega}) = \{ \zeta \in C(\bar{\Omega});\zeta = 0\ \text{on}\ \partial \Omega \} \]and\[C_0^2 (\bar{\Omega}) = \{ \zeta \in C^2 (\bar{\Omega});\zeta = 0\ \text{on}\ \partial \Omega \} \], and},
author = {H. Brezis and M. Marcus and A. C. Ponce},
booktitle = {Mathematical Aspects of Nonlinear Dispersive Equations (AM-163)},
pages = {55--110},
publisher = {Princeton University Press},
title = {Nonlinear Elliptic Equations with Measures Revisited},
year = {2007}
}