  In this paper, we study problem involving both local as well as nonlocal
operator, precisely the classical Laplace operator and the fractional Laplace
operator in the presence of mixed boundary conditions, that is
\begin{equation*} \label{1}
  \begin{cases} \tag{$P_\lambda$}
  \mathcal{L}u = \lambda u^{q} + u^{p}, \quad u>0 \quad
  \text{in }\Omega,
  u=0~~\text{in} ~~\Pi_1,
  \mathcal{N}_s(u)=0 ~~\text{in} ~~{\Pi_2},
  \frac{\partial u}{\partial \nu}=0 ~~\text{in}~~ \partial \Omega \cap
\overline{\Pi_2}.
  \end{cases}
  \end{equation*}
  where $\Omega \subseteq \mathbb{R}^n$ is a bounded domain with smooth
boundary, $\lambda >0$ is a real parameter, $ 0 < q < 1<p $, $n>2$ and
  $$\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1).$$
  Here $\Pi_{1}$, $\Pi_{2}$ are open subsets of $\mathbb{R}^n\setminus{\Omega
}$ such that $\overline{\Pi_{1}} \cup \overline{\Pi_{{2}}}=
\mathbb{R}^n\setminus{\Omega}$, $\Pi_{1} \cap \Pi_{{2}}= \emptyset.$ We
establish existence and nonexistence of weak solution, when $p<
\frac{n+2}{n-2}$ and multiplicity, when $p=\frac{n+2}{n-2}$ with respect to
parameter $\lambda$ for the above problem, using variational methods.
