  We study existence and multiplicity of nontrivial solutions of the following
problem $$ \left\{ \begin{array}{rcll} -\Delta_p u+(-\Delta_p)^{s} u & = &
\lambda|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }\Omega,\\ u & = & 0 & \mbox{
on } \mathbb{R}^{N} \setminus \Omega, \end{array} \right. $$ where
$\Omega\subset \mathbb{R}^N$ is a bounded open set with smooth boundary,
dimension $N\geq 2$, parameter $\lambda>0$, exponents $0<s<1<p<N$, while
$q\in(1,p^{\ast})$ with $p^{\ast}=\frac{Np}{N-p}$. The problem is driven by an
operator of mixed order obtained by the sum of the classical $p$-Laplacian and
of the fractional $p$-Laplacian. We analyze three different scenarios depending
on exponent $q$. For this, we combine variational methods with some topological
techniques, such as the Krasnoselskii genus and the Lusternik-Schnirelman
category theories.
