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

TL;DR

This paper investigates the existence, nonexistence, and multiplicity of positive solutions for a class of elliptic problems involving mixed operators and boundary conditions, with a focus on concave-convex nonlinearities and functional analysis tools.

Contribution

It introduces a functional framework for mixed boundary problems with fractional operators and nonlinearities, and provides new results on solution existence and properties.

Findings

Established conditions for solution existence and nonexistence.

Derived results related to Picone's identity and maximum principles.

Analyzed the impact of parameters on solution multiplicity.

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\: &= \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.