  In this work we analyze the asymptotic behavior of the solutions of the
$p$-Laplacian equation with homogeneous Neumann boundary conditions set in
bounded thin domains as
  $$R^\varepsilon=\left\lbrace(x,y)\in\mathbb{R}^2:x\in(0,1)\mbox{ and
}0<y<\varepsilon G\left(x,{x}/{\varepsilon}\right)\right\rbrace.$$
  We take a smooth function $G:(0,1)\times\mathbb{R} \mapsto \mathbb{R}$,
$L$-periodic in the second variable, which allows us to consider locally
periodic oscillations at the upper boundary. The thin domain situation is
established passing to the limit in the solutions as the positive parameter
$\varepsilon$ goes to zero.
