Existence of three positive solutions for a nonlocal singular dirichlet boundary problem

TL;DR

This paper proves the existence of at least three positive solutions for a nonlocal singular boundary value problem involving the fractional Laplacian, using sub-supersolutions and critical point theory.

Contribution

It establishes the existence of multiple solutions for a nonlocal singular problem, introducing a new existence result for an infinite semipositone nonlocal problem.

Findings

At least three positive solutions exist for certain parameters.

The method of sub-supersolutions is effective for nonlocal singular problems.

A new existence result for semipositone nonlocal problems is developed.

Abstract

In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem \begin{equation*} (P_\la)\left\{ \begin{split} (-\De)^su &= \la\frac{f(u)}{u^q}, \; \; u>0 \;\; \text{in}\;\; \Om,\\ u &= 0\;\; \text{in}\;\; \mb R^n \setminus \Om \end{split} \right. \end{equation*} where $(-\De)^s$ denotes the fractional Laplace operator for $s\in (0,1)$, $n>2s$, $q \in (0,1)$, $\la>0$ and $\Om$ is smooth bounded domain in $\mb R^n$. Here $f :[0,\infty) \to [0,\infty)$ is a continuous nondecreasing map satisfying $\lim\limits_{u\to \infty}\frac{f(u)}{u^{q+1}}=0$. We show that under certain additional assumptions on $f$, $(P_\la)$ possesses at least three distinct solutions for a certain range of $\la$. We use the method of sub-supersolutions and a critical point theorem by Amann \cite{amann} to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest. \medskip