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Quasilinear relaxed Dirichlet problems. (English) Zbl 0863.35041

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.

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
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