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

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.