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Classification of solutions for an integral equation. (English) Zbl 1093.45001
Commun. Pure Appl. Math. 59, No. 3, 330-343 (2006); corrigendum 59, No. 7, 1064 (2006).
The authors consider the integral equation $u\left( x\right) =\int_{{\mathbb R}^n}\frac 1{\left| x-y\right| ^{n-\alpha }}u\left( y\right) ^{\frac{n+\alpha }{n-\alpha }}dy$ which is equivalent to the semilinear partial differential equation $\left( -\Delta \right) ^{\alpha /2}u=u^{\frac{n+\alpha }{n-\alpha }}.$ Using the method of moving planes in an integral form, the authors prove that every positive regular solution $$u\left( x\right)$$ is radially symmetric and monotone about some point and therefore one shows that the only possible form of the solution that one can assume is the form $$\displaystyle c\left( \frac t{t^2+\left| x-x_0\right| ^2}\right) ^{\frac{ n-\alpha }2}.$$

##### MSC:
 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 45M20 Positive solutions of integral equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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