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} ~~{Î _2}, \frac{\partial u}{\partial ν}&=0 ~~\text{in}~~ \partial Ω\cap \overline{Î _2}. \end{split} \right.\tag{$P_{\hat{I}}»$} \end{equation*} {where $D=\left(\hat{I}\text{c◯}\cup {\hat{I}}_2\cup (\partial \hat{I}\text{c◯}\cap \overline{{\hat{I}}_2})\right)$ and $D^c$ is the complement of $D$, $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^n$ is a non empty open set, ${\hat{I}}_1$, ${\hat{I}}_2$ are open subsets of ${\mathbb{R}}^n\setminus \bar{\hat{I}}\text{c◯}$ such that $\overline{{\hat{I}}_1\cup {\hat{I}}_2}={\mathbb{R}}^n\setminus \hat{I}\text{c◯}$, ${\hat{I}}_1\cap {\hat{I}}_2=\varnothing$ and $\hat{I}\text{c◯}\cup {\hat{I}}_2$ is a bounded set with smooth boundary}, $\hat{I}»>0$ is a real parameter, $0<q<1<p$, $n>2$ and $\mathcal{L}=-\hat{I}”+(-\hat{I}”{)}^s,\text{for}s\in (0,1).$ 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 $\hat{I}»$, $q$ and $p$}. Our article also contains results related to Picone's identity, strong maximum principles and comparison principles.

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