A priori estimates and Liouville type theorems for semilinear equations and systems with fractional Laplacian

Abstract

In this paper, we firstly obtain a priori estimates of nonnegative solutions to fractional Laplacian equations and systems with critical Sobolev exponent. Secondly, by deriving the universal estimates of solutions, we establish the connections between Liouville type theorems and local properties of nonnegative solutions to fractional Laplacian equations and fractional Lane-Emden systems with subcritical Sobolev exponent. Our main results improve the work of Chen et al. (2016) [8] and Polácik et al. (2007) [26]. Specially, for fractional Lane-Emden systems, our results seem to be the first results on universal estimates to cover the full subcritical range. Furthermore, the novelty of our method lies in a completely new blow-up analysis coupled with a truncating technique and the construction of barrier function which is sufficiently different from the blow-up method used in [8,26].