A direct blowing-up and rescaling argument on nonlocal elliptic equations (Q2816974)

In this paper, the authors develop a direct blowing-up and rescaling argument for nonlinear equations involving nonlocal elliptic operators including the fractional Laplacian. The novelty of this paper is that the authors work directly on the nonlocal operator. Instead, in their seminal papers, Caffarelli and Silvestre used the conventional extension method to localise the problem. More precisely, the authors carry on a blowing-up and rescaling argument in order to obtain a priori estimates on the positive solutions. Based on this estimate and the Leray-Schauder degree theory, they prove the existence of positive solutions.