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