The purpose of this paper is to study the homogenization of the following problem \(\begin{aligned} \left\{ \begin{array}{lll} -\sum \limits _{i=0}^{N}\partial _i (a(\frac{x}{\epsilon })|\partial _{i}u_\epsilon |^{p_i-2}\partial _{i}u_\epsilon )=f_\epsilon & \text { in }\Omega , \\ u_\epsilon =0& \text { on }\partial {\Omega }, \end{array} \right. \end{aligned}\) where \(\Omega \) is a bounded open subset of \(\mathbb {R}^{N}\) with smooth boundary \(\partial {\Omega }\) , \(p_i > 1\) for all \(i = 1 \dots N\) , \(\epsilon \) is a small real parameter, and \(a(\cdot )\) is a \(Y\) -periodic function that satisfies suitable conditions. Without using two-scale convergence, we prove that \(u_\epsilon \) converges weakly to some \(u\) as \(\epsilon \rightarrow 0\) , where \(u\) is the solution of the homogenized problem. We also give some properties of the homogenized operator.
