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Let n be a positive integer and let 0 < α < n. In this paper, we continue our study of the integral equationWe mainly consider singular solutions in subcritical, critical, and super critical cases, and obtain qualitative properties, such as radial symmetry, monotonicity, and upper bounds for the solutions.

ABSTRACT. In this paper, we introduce a direct method of moving spheres for the nonlocal fractional Laplacian (−△) α/2 with 0 < α < 2, in which a key ingredient is the narrow region maximum principle. As immediate applications, we classify the non-negative solutions for a semilinear equation involving the fractional Laplacian in R n ; we prove a non-existence result for prescribing Q α curvature equation on S n ; then by combining the direct method of moving planes and moving spheres, we establish a Liouville type theorem on a half Euclidean space. We expect to see more applications of this method to many other nonlinear equations involving non-local operators.

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In this paper, we consider nonlinear equations involving the fractional p-LaplacianWe prove a maximum principle for anti-symmetric functions and obtain other key ingredients for carrying on the method of moving planes, such as a key boundary estimate lemma. Then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p-Laplacian in a unit ball and in the whole space. We believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators.Key words The fractional p-Laplacian, maximum principles for anti-symmetric functions, a key boundary estimate, method of moving planes, radial symmetry, monotonicity.

In this paper, we consider the critical order Hardy-Hénon equationswhere n ≥ 4 is even, −∞ < a < n, and 1 < p < +∞. We first prove a Liouville theorem (Theorem 1.1), that is, the unique nonnegative solution to this equation is u ≡ 0. Then as an immediate application, we derive a priori estimates and hence existence of positive solutions to critical order Lane-Emden equations in bounded domains (Theorem 1.4 and 1.5). Our results seem to be the first Liouville theorem, a priori estimates, and existence on the critical order equations in higher dimensions (n ≥ 3). Extensions to super-critical order Hardy-Hénon equations and inequalities will also be included (Theorem 1.7 and 1.9).

In this paper, we consider equations involving fully nonlinear nonlocal operatorsWe prove a maximum principle and obtain key ingredients for carrying on the method of moving planes, such as narrow region principle and decay at infinity. Then we establish radial symmetry and monotonicity for positive solutions to Dirichlet problems associated to such fully nonlinear fractional order equations in a unit ball and in the whole space, as well as non-existence of solutions on a half space. We believe that the methods develop here can be applied to a variety of problems involving fully nonlinear nonlocal operators.We also investigate the limit of this operator as α→2 and show that F α (u(x))→a(−△u(x)) + b|▽u(x)| 2 .

In this paper, we develop a direct 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 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.