Renormalized solutions to elliptic equations with measure data in unbounded domains. (English) Zbl 1221.35151
Summary: We consider the nonlinear elliptic problem
\[ -\text{div}(a(x,\nabla u))+a_0(x,u)=\mu \text{ in }\Omega\quad u=0\text{ on } \partial\Omega, \]
where \(\Omega\) is an open (possibly unbounded) subset of \(\mathbb R^N\), \(N\geq 2\), \(\mu\) is a Radon measure with bounded variation in \(\Omega\), and \((u)\mapsto-\text{div}(a(x,\nabla u))+a_0(x,u)\) is a monotone operator acting in \(W^{1,p}_0(\Omega)\), \(1<p\leq N\). We prove that for every \(\mu\) there exists at least a renormalized solution \(u\) to the problem, that is a distributional solution with additional summability properties. Moreover, if the operator is strictly monotone and \(\mu\) does not charge sets of capacity zero, such a solution is unique.
\[ -\text{div}(a(x,\nabla u))+a_0(x,u)=\mu \text{ in }\Omega\quad u=0\text{ on } \partial\Omega, \]
where \(\Omega\) is an open (possibly unbounded) subset of \(\mathbb R^N\), \(N\geq 2\), \(\mu\) is a Radon measure with bounded variation in \(\Omega\), and \((u)\mapsto-\text{div}(a(x,\nabla u))+a_0(x,u)\) is a monotone operator acting in \(W^{1,p}_0(\Omega)\), \(1<p\leq N\). We prove that for every \(\mu\) there exists at least a renormalized solution \(u\) to the problem, that is a distributional solution with additional summability properties. Moreover, if the operator is strictly monotone and \(\mu\) does not charge sets of capacity zero, such a solution is unique.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 35A15 | Variational methods applied to PDEs |
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