On the first eigenvalue of the generalized laplacian

In this work we investigate the energy of minimizers of Rayleigh-type quotients of the form ∫_Ω A(|∇ u|)\, dx/∫_Ω A(|u|)\, dx. These minimizers are eigenfunctions of the generalized laplacian defined as Δₐ u = div(a(|∇ u|)∇ u/|∇ u|) 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 Δ₂ condition is imposed. Since the problem is not homogeneous, the energy of minimizers is known to strongly depend on the normalization parameter α =∫_Ω A(|u|)\, dx. In this work we precisely analyze this dependence and show differentiability of the energy with respect to α and, moreover, the limits as α→ 0 and α→ ∞ of the Rayleigh quotient. The nonlocal version of this problem is also analyzed.