Uniform a priori estimates for solutions of higher critical order fractional equations

Abstract

In this paper, we establish uniform a priori estimates for positive solutions to higher critical order fractional Lame-Emden equations for all large exponents by adopting a new idea of using the combination of Green’s representations with suitable cut-off functions. We obtain two key ingredients needed in the estimates. One is the monotonicity of solutions in the inward normal direction near the boundary by using the method of moving planes in a local way, and the other is the integration by parts formula for higher order fractional operators. Both of them will become useful tools in the analysis of higher order fractional equations. It is well-known that, with such a priori estimate, one will be able to obtain the existence of solutions via topological degree or continuation arguments.