A property of nonlinear elliptic equations when the right-hand side is a measure
The authors consider the problem: \[ -\text{div} a(x,u,Du) = \mu \quad \text{in} \quad \Omega, \qquad u = 0 \quad \text{on} \quad \partial \Omega, \tag{1} \] where \(A(u) = - \text{div} a(x,u,Du)\) is a Leray-Lions operator satisfying natural growth conditions of \(p\)-type and \(\mu\) is a bounded Radon measure. A function \(u\) is a SOLA (solution obtained as limit of approximation) of (1) if \(u\) is a strong limit of \(u_ n\) in \(W^{1,q}_ 0 (\Omega)\) for each \(q < \overline q = N(p-1)/(N-1)\), and the functions \(u_ n \in W_ 0^{1,p} (\Omega)\) solve (1) with the right hand side measures \(\mu_ n\) \(L^{p'}\)-functions that converge to \(\mu\) weakly in the sense of measures. The main theorem states that for a SOLA \(u\) of (1) the truncations \(T_ k (u)\) of \(u\) at the height \(k\) solve (1) in the sense of distributions, i.e. \(A(T_ k(u))\) is a bounded Radon measure. Moreover, \(A(T_ k(u))\) tends to \(\mu\) weakly in the sense of measures as \(k\) tends to \(\infty\). In the case where \(\mu\) is a nonnegative measure and \(a(x,u,Du)\) does not depend on \(u\), the theorem can be found proven in [\textit{T. Kilpeläinen} and \textit{J. Malý}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 19, No. 4, 591-613 (1992; Zbl 0797.35052)].
solution obtained as limit of approximation
Leray-Lions operator satisfying natural growth conditions
scientific article; zbMATH DE number 650840