On the first eigenvalue of the generalized laplacian

Abstract

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 $a(t)=A^{\prime }(t)$ 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.

22 pages. Submitted