Eigenvalue homogenization problem with indefinite weights

TL;DR

This paper investigates the asymptotic behavior of eigenvalues in nonlinear elliptic equations with indefinite weights, revealing conditions under which eigenvalues diverge or converge, advancing understanding of homogenization in indefinite weight problems.

Contribution

It provides a detailed analysis of the asymptotic behavior of variational eigenvalues for nonlinear elliptic equations with sign-changing weights, including conditions for divergence and convergence.

Findings

Positive eigenvalues tend to infinity when the average weight is nonpositive.

Eigenvalues converge to those of the limit problem when the average weight is positive.

The study extends homogenization theory to problems with indefinite weights.

Abstract

In this work we study the homogenization problem for nonlinear elliptic equations involving $p-$Laplacian type operators with sign changing weights. We study the asymptotic behavior of variational eigenvalues, which consist on a double sequence of eigenvalues. We show that the $k-$th positive eigenvalue goes to infinity when the average of the weight is nonpositive, and converge to the $k-$th variational eigenvalue of the limit problem when the average is positive for any $k\ge 1$.