Finzi Vita, Stefano; Murat, François; Tchou, Nicoletta A. Quasilinear relaxed Dirichlet problems. (English) Zbl 0863.35041 SIAM J. Math. Anal. 27, No. 4, 977-996 (1996). This work is devoted to the study of quasilinear relaxed Dirichlet problems that can “formally” be written as \[ -\Delta u+\lambda_0u+\mu u= f(x,u,Du)\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(\lambda_0\geq 0\), \(f\) satisfies a quadratic growth condition with respect to \(Du\) and \(\mu\) is a nonnegative Borel measure in \(\Omega\) that vanishes on subsets with zero harmonic capacity. After giving a precise meaning to solutions of such a problem, two existence results are proved: the first one concerning bounded solutions and the second one related to the unbounded case. Finally, it is proved a stability property of solutions with respect to the \(\gamma\)-convergence of measures, when the growth conditions of \(f\) with respect to \(Du\) is strictly subquadratic. Reviewer: L.A.Fernandez (Santander) Cited in 2 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35B35 Stability in context of PDEs 35R05 PDEs with low regular coefficients and/or low regular data Keywords:harmonic capacity; existence; \(\gamma\)-convergence of measures PDFBibTeX XMLCite \textit{S. Finzi Vita} et al., SIAM J. Math. Anal. 27, No. 4, 977--996 (1996; Zbl 0863.35041) Full Text: DOI