Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction

TL;DR

This paper investigates the asymptotic behavior of eigenfunctions and eigenvalues of the p-Laplace operator in cylindrical domains as their length tends to infinity, extending previous linear case results to nonlinear scenarios.

Contribution

It generalizes earlier linear results to the nonlinear p-Laplace operator, analyzing eigenfunction and eigenvalue asymptotics in unbounded cylindrical domains.

Findings

Asymptotic behavior characterized for the first eigenfunction in nonlinear case.

Higher eigenvalues' asymptotics established for linear case.

Second eigenvalues analyzed using topological degree methods.

Abstract

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et.al. "Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity" published in Asymptotic Analysis in 2013, where the linear case p=2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.