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Exact solutions of nonlinear conformally invariant integral equations in $\bold R^3$. (English) Zbl 1073.45003
A necessary condition for an integral equation $$u(x)= {1\over 8\pi} \int_{\bbfR^3}|x- y|u^{-q}(y)\,dy\quad\text{in }\bbfR^3\qquad (q> 0)$$ appearing in conformal geometry to have a $C^4$ entire positive solution is that $q= 7$. If it is the case then this solution is of the form $u(x)= c(1+|x|^2)^{1/2}$ up to dilation and translations (where $c$ is a constant).

45G05Singular nonlinear integral equations
45M20Positive solutions of integral equations
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
Full Text: DOI
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