On the Neumann $(p,q)$-eigenvalue problem in Hölder singular domains

Abstract

In the article we study the Neumann $(p,q)$-eigenvalue problems in bounded Hölder ${\hat{I}}^3$-singular domains $\hat{I}{\text{c◯}}_{\hat{I}}^3\subset {\mathbb{R}}^n$. In the case $1<p<\infty$ and $1<q<p_{\hat{I}}^3$ 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.

15 pages