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Nonlinear elliptic equations with right hand side measures. (English) Zbl 0812.35043
Let \(\Omega\subset \mathbb{R}^ N\) be a bounded open set, \(p\in (2- 1/n,N]\), \(\alpha,\beta>0\), \(h\in L^{p'} (\Omega)\) and \(a: \Omega\times \mathbb{R}\times \mathbb{R}^ N\to \mathbb{R}^ N\) a Carathéodory function such that \(a(x,s,\xi)\geq \alpha| \xi|^ 2\), \(a(x,s, \xi)\geq \beta[h(x)+ | s|^{p-1}+ |\xi |^{p-1}]\) and \([a(x,s, \xi)- a(x,s,\gamma)] (\xi-\gamma)>0\) for all \(s\in\mathbb{R}\), \(\xi,\gamma\in \mathbb{R}^ N\) with \(\xi\neq\gamma\) and for almost all \(x\in\Omega\). The authors prove the following results about existence and regularity of solutions of the variational problem \[ \int_ \Omega a(x,u, Du)Dv dx= \int_ \Omega fv dx \quad \text{for all } v\in C_ 0^ \infty (\Omega), \quad a(x,u,Du)\in L^ 1(\Omega). \] If \(f\) is a bounded Radon measure on \(\Omega\) then there exists a solution \(u\), and \(u\in W_ 0^{1,q} (\Omega)\) for any \(q< \overline{q}:= N(p-1)/ (N-1)\). Moreover, if \(| f|\log | f|\in L^ 1(\Omega)\) (resp. if \(f\in L^ m(\Omega)\) with \(1<m< Np/ (Np- N+p)\)) then \(u\in W^{1, \overline{q}} (\Omega)\) (resp. \(u\in W^{1, (p-1) m^*} (\Omega)\) with \(m^*:= m/(m- p)\)).
Reviewer: L.Recke (Berlin)

35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data
Full Text: DOI
[1] DOI: 10.1016/0022-1236(89)90005-0 · Zbl 0707.35060 · doi:10.1016/0022-1236(89)90005-0
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[5] DOI: 10.1007/BF01766148 · Zbl 0687.35042 · doi:10.1007/BF01766148
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