Let n be a positive integer and let 0 < α < n. Consider the integral equationWe prove that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the formwith some constant c = c(n, α) and for some t > 0 and x 0 ∈ R n . This solves an open problem posed by Lieb [12]. The technique we use is the method of moving planes in an integral form, which is quite different from those for differential equations. From the point of view of general methodology, this is another interesting part of the paper.Moreover, we show that the family of well-known semilinear partial differential equations (− ) α/2 u = u n+α n−α is equivalent to our integral equation (0.1), and we thus classify all the solutions of the PDEs.

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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.

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