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Classification of solutions for an integral equation. (English) Zbl 1093.45001
The authors consider the integral equation $$ u\left( x\right) =\int_{{\Bbb R}^n}\frac 1{\left\vert x-y\right\vert ^{n-\alpha }}u\left( y\right) ^{\frac{n+\alpha }{n-\alpha }}dy $$ 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\vert x-x_0\right\vert ^2}\right) ^{\frac{ n-\alpha }2}.$

45E10Integral equations of the convolution type
45M20Positive solutions of integral equations
35J65Nonlinear boundary value problems for linear elliptic equations
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