Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian

In this paper we consider the existence and uniqueness of weak positive solution for nonlocal equations of the square root of the Laplacian with singular nonlinearity. The remarkable feature of this paper is the fact that the natural associated functional fails to be Fr´echet differentiable, critical point theory could not be applied to obtain the existence of weak positive solution. We first establish the priori estimate of weak solution of approximating problems. Then the weak positive solution is constructed by combining sub-and supersolutions method and truncate technology.