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In this paper, we develop a direct {\em blowing-up and rescaling} argumentfor a nonlinear equation involving the fractional Laplacian operator. Insteadof using the conventional extension method introduced by Caffarelli andSilvestre, we work directly on the nonlocal operator. Using the integraldefining the nonlocal elliptic operator, by an elementary approach, we carry ona {\em blowing-up and rescaling} argument directly on nonlocal equations andthus obtain a priori estimates on the positive solutions for a semi-linearequation involving the fractional Laplacian.  We believe that the ideas introduced here can be applied to problemsinvolving more general nonlocal operators.

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