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 \(\gamma \) $\gamma$ -singular domains \(\Omega _{\gamma }\subset {\mathbb {R}}^n\) ${\mathrm{\Omega }}_{\gamma }\subset {\mathbb{R}}^n$ . In the case \(1<p<\infty \) $1<p<\infty$ and \(1<q<p^{*}_{\gamma }\) $1<q<p_{\gamma }^{\ast }$ 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.