A new concept of reduced measure for nonlinear elliptic equations

Abstract

We study the existence of solutions of the nonlinear problem (i) − 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 ( t ) = 0 , ∀ t ⩽ 0 . Given g, Eq. (i) need not have a solution for every measure μ, and we say that μ is a good measure if (i) admits a solution. We show that for every μ there exists a largest good measure μ * ⩽ μ . This reduced measure μ * has a number of remarkable properties. To cite this article: H. Brezis et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).