On elliptic problems with mixed operators and Dirichlet-Neumann boundary conditions

In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by \begin{equation*} \label{1} \left\{\begin{split} \mathcal{L}u\: &= \lambda u^{q} + u^{p}, \quad u>0 ~~ \text{in} ~\Omega, u&=0~~\text{in} ~~{D^c}, \mathcal{N}_s(u)&=0 ~~\text{in} ~~{\Pi_2}, \frac{\partial u}{\partial \nu}&=0 ~~\text{in}~~ \partial \Omega \cap \overline{\Pi_2}. \end{split} \right.\tag{$P_\lambda$} \end{equation*} {where $D= \left(\Omega \cup {\Pi_2} \cup (\partial\Omega\cap\overline{\Pi_2})\right)$ and $D^c$ is the complement of $D$, $\Omega \subseteq \mathbb{R}^n$ is a non empty open set, $\Pi_{1}$, $\Pi_{2}$ are open subsets of $\mathbb{R}^n\setminus{\bar \Omega }$ such that $\overline{{\Pi_{1}} \cup {\Pi_{{2}}}}= \mathbb{R}^n\setminus{\Omega}$, $\Pi_{1} \cap \Pi_{{2}}= \emptyset$ and $\Omega\cup \Pi_2$ is a bounded set with smooth boundary}, $\lambda >0$ is a real parameter, $ 0 < q < 1<p $, $n>2$ and $\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1).$ We first present a functional setting to study any problem involving $\mathcal L$ under mixed boundary conditions in the presence of concave-convex power nonlinearity, {for a suitable range of $\lambda$, $q$ and $p$}. Our article also contains results related to Picone's identity, strong maximum principles and comparison principles.