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: &= λu^{q} + u^{p}, \quad u>0 ~~ $\text{in}$ ~Ω, u&=0~~$\text{in}$ ~~{D^c}, $\mathcal{N}$_s(u)&=0 ~~$\text{in}$ ~~{Π_2}, \frac{\partial u}{\partial ν}&=0 $\text{in}$ \partial Ω\cap \overline{Π_2}. \end{split} \right.\tag{$P_{\lambda }$} \end{equation*} {where $D=(\mathrm{\Omega }\cup {\mathrm{\Pi }}_2\cup (\mathrm{\partial }\mathrm{\Omega }\cap \overline{{\mathrm{\Pi }}_2}))$ and $D^c$ is the complement of $D$, $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ is a non empty open set, ${\mathrm{\Pi }}_1$, ${\mathrm{\Pi }}_2$ are open subsets of ${\mathbb{R}}^n\setminus \bar{\mathrm{\Omega }}$ such that $\overline{{\mathrm{\Pi }}_1\cup {\mathrm{\Pi }}_2}={\mathbb{R}}^n\setminus \mathrm{\Omega }$, ${\mathrm{\Pi }}_1\cap {\mathrm{\Pi }}_2=\mathrm{\varnothing }$ and $\mathrm{\Omega }\cup {\mathrm{\Pi }}_2$ is a bounded set with smooth boundary}, $\lambda >0$ is a real parameter, $0<q<1<p$, $n>2$ and $\mathcal{L}=-\mathrm{\Delta }+(-\mathrm{\Delta }{)}^s,\text{ for }s\in (0,1)\mathrm{.}$ 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.