Homogenization of quasi-linear equations with natural growth terms

Abstract

In this paper we deal with the limit behaviour of the bounded solutions u_ε of quasi-linear equations of the form$- div\left( {a\left( {\tfrac{x}{\varepsilon },Du_\varepsilon } \right)} \right) + \gamma \left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon = H\left( {\tfrac{x}{\varepsilon },u_\varepsilon ,Du_\varepsilon } \right) + h(x)$ of Ω with Dirichlet boundary conditions on σΩ. The map a=a(x,ϕ) is periodic in x, monotone in ϕ, and satisfies suitable coerciveness and growth conditions. The function H=H(x,s,ϕ) is assumed to be periodic in x, continuous in [s,ϕ] and to grow at most like |ξ|^p. Under these assumptions on a and H we prove that there exists a function H⁰=H⁰(s,ϕ) with the same behaviour of H, such that, up to a subsequence, (u_ε) converges to a solution u of the homogenized problem -div(b(Du)) + γ|u|^p-2u = H⁰(u,Du) + h(x) on Ω, where b depends only on a and has analogous qualitative properties.