Mixed local and nonlocal Dirichlet $(p,q)$-eigenvalue problem

In this article, we consider the spectral problem for the mixed local and nonlocal $p$-Laplace operator. We discuss the existence and regularity of eigenfunction of the associated Dirichlet $(p,q)$-eigenvalue problem in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ under the assumption that $1<p<\mathrm{\infty }$ and $1<q<p^{\ast }$ where $p^{\ast }=\frac{Np}{N-p}$ if $1<p<N$ and $p^{\ast }=\mathrm{\infty }$ if $p\ge N$.