Mixed local-nonlocal quasilinear problems with critical nonlinearities

We study existence and multiplicity of nontrivial solutions of the following problem $$\{\begin{array}{rcll}-{\mathrm{\Delta }}_{p}u+(-{\mathrm{\Delta }}_p{)}^{s}u& =& \lambda \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{p^{\ast }-2}u& in\mathrm{\Omega },\\ u& =& 0& on{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded open set with smooth boundary, dimension $N\ge 2$, 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.