On some Variational Problems set on domains tending to infinity

Let $Ω_\ell = \ellω_1 \times ω_2$ where $ω_1 \subset \R^p$ and $ω_2 \subset \R^{n-p}$ are assumed to be open and bounded. We consider the following minimization problem: $E_{Ω_\ell}(u_\ell) = \min_{u\in W_0^{1,q}(Ω_\ell)}E_{Ω_\ell}(u)$ where $E_{Ω_\ell}(u) = \int_{Ω_\ell}F(\grad u)-fu$, $F$ is a convex function and $f\in L^{q'}(ω_2)$. We are interested in studying the asymptotic behavior of the solution $u_\ell$ as $\ell$ tends to infinity.

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