# On the Second Eigenvalue of Combination Between Local and Nonlocal $p$-Laplacian

## TL;DR

This paper investigates the second eigenvalue of a combined local and nonlocal p-Laplacian operator using mountain pass characterization and explores related shape optimization problems.

## Contribution

It introduces a novel approach to characterize the second eigenvalue of the combined operator and addresses shape optimization issues associated with these eigenvalues.

## Key findings

- Characterization of the second eigenvalue via mountain pass theorem
- Analysis of shape optimization problems for the combined operator
- Insights into the spectral properties of local and nonlocal p-Laplacians

## Abstract

In this paper, we study Mountain Pass Characterization of the second eigenvalue of the operator $-\De_p u -\De_{J,p}u$ and study shape optimization problems related to these eigenvalues.