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

In the article we study the Neumann $(p,q)$-eigenvalue problems in bounded Hölder $\gamma$-singular domains ${\mathrm{\Omega }}_{\gamma }\subset {\mathbb{R}}^n$. In the case $1<p<\mathrm{\infty }$ and $1<q<p_{\gamma }^{\ast }$ 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.