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Some properties of reachable solutions of nonlinear elliptic equations with measure data. (English) Zbl 1033.35034

Summary: Let \({\mathcal A}: W^{1,p}_0(\Omega)\to W^{-1,q}(\Omega)\), \(1/p+ 1/q= 1\), be a monotone operator of the form \({\mathcal A}(u)= -\text{div}(A(x, Du))\) on a bounded open set \(\Omega\) of \(\mathbb{R}^N\), \(N\geq 2\). Given a measure \(\mu\) with bounded variation on \(\Omega\) and a function \(F\in L^q(\Omega, \mathbb{R}^N)\), we study some properties of those solutions of the equation \({\mathcal A}(u)= \mu-\text{div}(F)\) which can be approximated by solutions \(u_n\) of equations of the form \({\mathcal A}(u_n)= f_n- \text{div}(F_n)\), where \(f_n\) are functions in \(C^\infty_c(\Omega)\) converging to \(\mu\) in the weak\(^*\) topology of measures, and \(F_n\) are functions in \(C^\infty_c(\Omega, \mathbb{R}^N)\) converging to \(F\) strongly in \(L^q(\Omega, \mathbb{R}^N)\).

MSC:

35J60 Nonlinear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data

Keywords:

Radon measures
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References:

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