H convergence for quasi-linear elliptic equations with quadratic growth

We consider in this paper the limit behavior of the solutionsu ^? of the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + \gamma u^\varepsilon = H^\varepsilon (x, u^\varepsilon , Du^\varepsilon ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ) \cap L^\infty (\Omega ), \hfill \\ \end{gathered}$$ whereH ^? has quadratic growth inDu ^? anda ^? (x) is a family of matrices satisfying the general assumptions of abstract homogenization. We also consider the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) = f \in H^{ - 1} (\Omega ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ), G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ), u^\varepsilon G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ) \hfill \\ \end{gathered}$$ whereG ^? has quadratic growth inDu ^? and satisfiesG ^? (x, s, ξ)s ≥ 0. Note that in this last modelu ^? is in general unbounded, which gives extra difficulties for the homogenization process. In both cases we pass to the limit and obtain an homogenized equation having the same structure.