## 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 $$(*)$$.

### MSC:

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

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