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Positive regular solutions to a singular integral equation. (English) Zbl 1183.45001

A positive regular solution \(u(x)\) of the singular integral equation
\[ u(x) = \int_{R^n} |x-y|^{\alpha-n}(u(y))^{(n+\alpha)/(n-\alpha)}dy, \quad 0 < \alpha < n, \tag{1} \]
is investigated. Under some conditions the radial symmetry and monotonicity of positive regular solutions of equation (1) are proved. The form of solutions is investigated, too. Proofs of the results are based on the method of moving planes.

MSC:

45G05 Singular nonlinear integral equations
45M20 Positive solutions of integral equations
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