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

Abstract

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{$P_λ$} \end{equation*} {where $D= \left(Ω\cup {Π_2} \cup (\partialΩ\cap\overline{Π_2})\right)$ and $D^c$ is the complement of $D$, $Ω\subseteq \mathbb{R}^n$ is a non empty open set, $Π_{1}$, $Π_{2}$ are open subsets of $\mathbb{R}^n\setminus{\bar Ω}$ such that $\overline{Π_{1} \cup {Π_{2}}}= \mathbb{R}^n\setminusΩ$, $Π_{1} \cap Π_{2}= \emptyset$ and $Ω\cup Π_2$ is a bounded set with smooth boundary}, $λ>0$ is a real parameter, $ 0 < q < 1 2$ and $\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$ 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 $λ$, $q$ and $p$}. Our article also contains results related to Picone's identity, strong maximum principles and comparison principles.