Minimization of quotients with variable exponents

Let Ω be a bounded domain of RN, p∈C1(Ω‾), q∈C(Ω‾) and l,j∈N. We describe the asymptotic behavior of the minimizers of the Rayleigh quotient ‖∇u‖lp(x)‖u‖jq(x), first when j→∞ and after when l→∞.