A large class of nonlocal elliptic equations with singular nonlinearities

TL;DR

This paper develops a unified theory for a broad class of nonlocal elliptic equations with singular nonlinearities, analyzing existence, uniqueness, and boundary behavior of solutions across various fractional operators.

Contribution

The work introduces a new approach to study nonlocal elliptic equations with singular nonlinearities, including multiple operator types and nonlinearities, providing a comprehensive classification of solutions.

Findings

Existence and uniqueness of solutions established.

Identification of two critical exponents for solution classification.

Boundary behavior characterized for different nonlinearities.

Abstract

In this work, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of equation $\mathbb{L}_\gamma^s u = \mathcal{F}(u)$, posed in a $C^2$ bounded domain $\Omega \subset \mathbb{R}^N$, with appropriate homogeneous boundary or exterior Dirichlet conditions. The operator $\mathbb{L}_\gamma^s$ belongs to a general class of nonlocal operators including typical fractional Laplacians such as restricted fractional Laplacian, censored fractional Laplacian and spectral fractional Laplacian. The nonlinear term $\mathcal{F}(u)$ covers three different amalgamation of nonlinearities: a purely singular nonlinearity $\mathcal{F}(u) = u^{-q}$ ($q>0$), a singular nonlinearity with a source term $\mathcal{F}(u) = u^{-q} + f(u)$, and a singular nonlinearity with an absorption term $\mathcal{F}(u) = u^{-q}-g(u)$. Based on a delicate analysis of the Green kernel associated to $\mathbb{L}_\gamma^s$, we develop a new unifying approach that empowered us to construct a theory for equation $\mathbb{L}_\gamma^s u = \mathcal{F}(u)$. In particular, we show the existence of two critical exponents $q^{\ast}_{s, \gamma}$ and $q^{\ast \ast}_{s, \gamma}$ which provides a fairly complete classification of the weak-dual solutions via their boundary behavior. Various types of nonlocal operators are discussed to exemplify the wide applicability of our theory.