On the first eigenvalue of the generalized laplacian

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.