Minimization of quotients with variable exponents

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, $p\in C^{1}(\overline{\Omega}),$ $q\in C(\overline{\Omega})$ and $l,j\in\mathbb{N}.$ We describe the asymptotic behavior of the minimizers of the Rayleigh quotient $\frac{\Vert\nabla u\Vert_{lp(x)}}{\Vert u\Vert_{jq(x)}}$, first when $j\rightarrow\infty$ and after when $l\rightarrow\infty.$