On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded

TL;DR

This paper investigates how the eigenvalues of nonlinear elliptic problems behave as the domain size grows unbounded, revealing differences between Dirichlet and mixed boundary conditions using Picone identity.

Contribution

It introduces a novel analysis of eigenvalue asymptotics for nonlinear elliptic problems on unbounded domains, highlighting boundary condition effects and simplifying proofs with Picone identity.

Findings

Eigenvalues exhibit distinct asymptotic behavior under Dirichlet and mixed boundary conditions.

Picone identity simplifies the analysis of nonlinear eigenvalue problems.

Asymptotic behavior differs significantly between boundary condition types for certain problems.

Abstract

We analyze the asymptotic behavior of the eigenvalues of nonlinear elliptic problems under Dirichlet boundary conditions and mixed (Dirichlet, Neumann) boundary conditions on domains becoming unbounded. We make intensive use of Picone identity to overcome nonlinearity complications. Altogether the use of Picone identity makes the proof easier with respect to the known proof in the linear case. Surprisingly the asymptotic behavior under mixed boundary conditions critically differs from the case of pure Dirichlet boundary conditions for some class of problems.