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Uniqueness theorem for integral equations and its application. (English) Zbl 1153.45005
The object of the paper is to study the existence of regular solutions of the following integral equation $$u(x)= \int_{\bbfR^n} |x-y|^p u^q(y)\,dy,\tag1$$ where $p$ and $q$ are real parameters. The main result of the paper states that if $p> 0$, then (1) has a $C^1$ positive solution if and only if $pq+ p+ 2n= 0$. This solution can be expressed by the formula $u(x)= a(b^2+ |x-x_0|^2)^{p/2}$. This result answers the question raised by {\it Y. Li} [J. Eur. Math. Soc. (JEMS) 6, No. 2, 153--180 (2004; Zbl 1075.45006)]. Some other results concerning (1) are also proved.

MSC:
45G05Singular nonlinear integral equations
45M20Positive solutions of integral equations
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References:
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