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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

$u\left(x\right)={\int }_{{ℝ}^{n}}{|x-y|}^{p}{u}^{q}\left(y\right)\phantom{\rule{0.166667em}{0ex}}dy,\phantom{\rule{2.em}{0ex}}\left(1\right)$

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 $u\left(x\right)=a\left({b}^{2}+|x-{x}_{0}{{|}^{2}\right)}^{p/2}$. This result answers the question raised by 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:
 45G05 Singular nonlinear integral equations 45M20 Positive solutions of integral equations