On some Variational Problems set on domains tending to infinity

Let ${\mathrm{\Omega }}_{\mathrm{\ell }}=\mathrm{\ell }{\omega }_1\times {\omega }_2$ where $\omega_1 \subset \R^p$and$\omega_2 \subset \R^{n-p}$ are assumed to be open and bounded. We consider the following minimization problem: $$E_{{\mathrm{\Omega }}_{\mathrm{\ell }}}(u_{\mathrm{\ell }})=\underset{u\in W_0^{1,q}({\mathrm{\Omega }}_{\mathrm{\ell }})}{\min }{E}_{{\mathrm{\Omega }}_{\mathrm{\ell }}}(u)$$ where $E_{{\mathrm{\Omega }}_{\mathrm{\ell }}}(u)={\int }_{{\mathrm{\Omega }}_{\mathrm{\ell }}}F(\text{grad}u)-fu$, $F$ is a convex function and $f\in L^{q^{\mathrm{\prime }}}({\omega }_2)$. We are interested in studying the asymptotic behavior of the solution $u_{\mathrm{\ell }}$ as $\mathrm{\ell }$ tends to infinity.