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Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. (English) Zbl 0857.35126

Summary: We consider the differential problem

A(u)=μinΩ,u=0onΩ,(*)

where Ω is a bounded, open subset of N , N2, A is a monotone operator acting on W 0 1,p (Ω), p>1, and μ is a Radon measure on Ω that does not charge the sets of zero p-capacity. We prove a decomposition theorem for these measures (more precisely, as the sum of a function in L 1 (Ω) and of a measure in W -1,p ' (Ω)), and an existence and uniqueness result for the so-called entropy solutions of (*).

MSC:
35R05PDEs with discontinuous coefficients or data
47H05Monotone operators (with respect to duality) and generalizations
35J60Nonlinear elliptic equations