Classification of solutions for an integral equation. (English) Zbl 1093.45001

Chen, Wenxiong; Li, Congming; Ou, Biao

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 \( \displaystyle c\left( \frac t{t^2+\left| x-x_0\right| ^2}\right) ^{\frac{ n-\alpha }2}.\)

Reviewer: Adrian Carabineanu (Bucureşti)

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

Keywords:

Integral equation; moving planes method; semilinear partial differential equations; radially symmetric solution; monotone solution; positive regular solution

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