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In this work, we address the questions of existence, uniqueness, and boundarybehavior 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 Dirichletconditions. The operator $\mathbb{L}_\gamma^s$ belongs to a general class ofnonlocal operators including typical fractional Laplacians such as restrictedfractional Laplacian, censored fractional Laplacian and spectral fractionalLaplacian. The nonlinear term $\mathcal{F}(u)$ covers three differentamalgamation 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 Greenkernel associated to $\mathbb{L}_\gamma^s$, we develop a new unifying approachthat 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 fairlycomplete classification of the weak-dual solutions via their boundary behavior.Various types of nonlocal operators are discussed to exemplify the wideapplicability of our theory.

A large class of nonlocal elliptic equations with singular nonlinearities