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.