On the Neumann $(p,q)$-eigenvalue problem in H\"older singular domains

TL;DR

This paper investigates the Neumann $(p,q)$-eigenvalue problem in bounded H"older singular domains, proving solvability, existence of minimizers, regularity of eigenfunctions, and eigenvalue estimates.

Contribution

It establishes the solvability and regularity results for the Neumann $(p,q)$-eigenvalue problem in H"older singular domains, which was previously unexplored.

Findings

Proved solvability of the eigenvalue problem.

Established existence of minimizers.

Derived estimates for eigenvalues.

Abstract

In the article we study the Neumann $(p,q)$-eigenvalue problems in bounded H\"older $\gamma$-singular domains $\Omega_{\gamma}\subset \mathbb{R}^n$. In the case $1<p<\infty$ and $1<q<p^{*}_{\gamma}$ we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some regularity results of the eigenfunctions and some estimates of $(p,q)$-eigenvalues.