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

作者：Chen, Wenxiong^*; Li, Congming; Li, Yan

来源：International Journal of Mathematics, 2016, 27(8): 1650064.

DOI：10.1142/S0129167X16500646

• Nonlocal elliptic operators
• fractional Laplacian
• blowing-up
• rescaling
• a priori estimates
• existence of solutions

In this paper, we develop a direct blowing-up and rescaling argument for nonlinear equations involving nonlocal elliptic operators including the fractional Laplacian. Instead of using the conventional extension method introduced by Caffarelli and Silvestre to localize the problem, we work directly on the nonlocal operator. Using the defining integral, by an elementary approach, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Based on this estimate and the Leray-Schauder degree theory, we establish the existence of positive solutions. We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators.

• 出版日期2016-7
• 单位西北工业大学