A large class of nonlocal elliptic equations with singular 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 $q_{s,\gamma }^{\ast }$ and $q_{s,\gamma }^{\ast \ast }$ 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.

41 Pages