  In this work we investigate the energy of minimizers of Rayleigh-type
quotients of the form $$ \frac{\int_\Omega A(|\nabla u|)\, dx}{\int_\Omega
A(|u|)\, dx}. $$
  These minimizers are eigenfunctions of the generalized laplacian defined as
$\Delta_a u = \text{div}\left(a(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)$
where $a(t)=A'(t)$ and the Rayleigh quotient is comparable to the associated
eigenvalue. On the function $A$ we only assume that it is a Young function but
no $\Delta_2$ condition is imposed.
  Since the problem is not homogeneous, the energy of minimizers is known to
strongly depend on the normalization parameter $\alpha =\int_\Omega A(|u|)\,
dx$. In this work we precisely analyze this dependence and show
differentiability of the energy with respect to $\alpha$ and, moreover, the
limits as $\alpha\to 0$ and $\alpha\to \infty$ of the Rayleigh quotient.
  The nonlocal version of this problem is also analyzed.
