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)=\mu\quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega,\tag{\(*\)} \] where \(\Omega\) is a bounded, open subset of \(\mathbb{R}^N\), \(N\geq 2\), \(A\) is a monotone operator acting on \(W_0^{1,p}(\Omega)\), \(p>1\), and \(\mu\) is a Radon measure on \(\Omega\) 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 \(L^1(\Omega)\) and of a measure in \(W^{-1,p'}(\Omega)\)), and an existence and uniqueness result for the so-called entropy solutions of \((*)\).


35R05 PDEs with low regular coefficients and/or low regular data
47H05 Monotone operators and generalizations
35J60 Nonlinear elliptic equations
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