Existence and uniqueness of positive solutions to fractional Laplacians with singular nonlinearities

Abstract

In this paper we prove the existence and uniqueness of positive classical solutions of fractional Laplacians with singular nonlinearities in a smooth bounded domain with zero Dirichlet boundary conditions. By the sub-supersolution method, we derive the existence of positive classical solutions to the approximation problems. In order to obtain the regularity, we first establish the existence of weak solutions for the fraction Laplacians. Thanks to Xia and Yang (2013), the regularity follows from the boundedness of weak solutions.