An Eigenvalue Problem with variable exponents

TL;DR

This paper investigates a complex nonlinear eigenvalue problem in a variable exponent Sobolev space, deriving the Euler-Lagrange equation, analyzing the asymptotic case, and proving local uniqueness of viscosity solutions.

Contribution

It introduces a novel eigenvalue problem with variable exponents, deriving its Euler-Lagrange equation and analyzing the asymptotic behavior with a new uniqueness result.

Findings

Derived Euler-Lagrange equation for the problem

Analyzed asymptotic case with variable infinity

Proved local uniqueness of viscosity solutions

Abstract

A highly nonlinear eigenvalue problem is studied in a Sobolev space with variable exponent. The Euler-Lagrange equation for the minimization of a Rayleigh quotient of two Luxemburg norms is derived. The asymptotic case with a "variable infinity" is treated. Local uniqueness is proved for the viscosity solutions.