An optimization problem for the first weighted eigenvalue problem plus a potential

TL;DR

This paper investigates the minimization of the first eigenvalue of a weighted p-Laplacian with potential, allowing the potential and weight to vary within rearrangement classes, extending previous results.

Contribution

It generalizes earlier work by analyzing the first eigenvalue minimization problem with variable potential and weight in rearrangement classes.

Findings

Extended previous eigenvalue minimization results

Characterized optimal potentials and weights within rearrangement classes

Provided new bounds and properties for the first eigenvalue

Abstract

In this paper, we study the problem of minimizing the first eigenvalue of the $p-$Laplacian plus a potential with weights, when the potential and the weight are allowed to vary in the class of rearrangements of a given fixed potential $V_0$ and weight $g_0$. Our results generalized those obtained in [9] and [5].