A direct blowing-up and rescaling argument on the fractional Laplacian equation

2015/06/01 by Wen‐Xiong Chen, Congming Li, Chen, Wenxiong +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1506.00582

openalex publication_date 2015/06/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Regularity of the obstacle problem for a fractional power of the laplace operator
• Uniqueness of Radial Solutions for the Fractional Laplacian

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