# Minimization of quotients with variable exponents

## TL;DR

This paper investigates the asymptotic behavior of minimizers for a variable exponent quotient involving gradient and function norms, as the exponents tend to infinity, providing insights into their limiting properties.

## Contribution

It characterizes the asymptotic behavior of minimizers of a variable exponent quotient as the exponents approach infinity, extending classical results to variable exponent spaces.

## Key findings

- Describes asymptotic behavior of minimizers as exponents go to infinity.
- Provides new insights into variable exponent Sobolev spaces.
- Extends classical quotient minimization results to variable exponents.

## Abstract

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, $p\in C^{1}(\overline{\Omega}),$ $q\in C(\overline{\Omega})$ and $l,j\in\mathbb{N}.$ We describe the asymptotic behavior of the minimizers of the Rayleigh quotient $\frac{\Vert\nabla u\Vert_{lp(x)}}{\Vert u\Vert_{jq(x)}}$, first when $j\rightarrow\infty$ and after when $l\rightarrow\infty.$