  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 $\Omega\subset\mathbb{R}^N$ under the assumption that
$1<p<\infty$ and $1<q<p^{*}$ where $p^{*}=\frac{N p}{N-p}$ if $1<p<N$ and
$p^{*}=\infty$ if $p\geq N$.
