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Differential Integral Equations 29 Citations From References: 2 From Reviews: 0 MR3513585 45G10 35B38 35J30 35J75 35J91 58J05 Zhu, Meijun [Zhu, Meijun1] (1-OK) Prescribing integral curvature equation. (English summary) Differential Integral Equations 29 (2016), no. 9-10, 889{904. n Given any positive, continuous, and antipodally symmetric function R on S , it is proved bαc−n n in this paper that there exists a positive, antipodally symmetric solution u 2 C (S ) to the following integral equation: Z α−n n+α (1) u(ξ) = R(η)jξ − ηj u(η) n−α dση; n S where α > n. Here, jξ − ηj denotes the chordal distance between ξ and η in Rn+1 and n by antipodally symmetric functions f we mean those f satisfying f(η) = f(−η) on S . Clearly (1) can be thought of as an integral version of the differential equation α/2 −1 n+α (2) (−∆) v = (R ◦ π )v n−α n n n on R obtained via the stereographic projection π: S ! R . However, (1) and (2) are not equivalent in the case α > n [see Y. S. Choi and X. Xu, J. Differential Equations 246 (2009), no. 1, 216{234; MR2467021; I. A. Guerra, J. Differential Equations 253 (2012), no. 11, 3147{3157; MR2968196; Trinh Viet Duoc and Qu^oc-AnhNg^o,\A´ note on radial 2 −q 3 solutions of ∆ u + u = 0 in R with exactly quadratic growth at infinity", preprint, arXiv:1511.09171]. To obtain such an existence result for (1), the author seeks critical points of the energy functional ZZ Z n+α α−n 2n n Jα,R(f) = R(ξ)R(η)f(ξ)f(η)jξ − ηj dσξdση f(η) n+α R(η)dση n n n S ×S S n on S . By using a reversed Hardy-Littlewood-Sobolev inequality recently proved by the author and J. B. Dou [see Int. Math. Res. Not. IMRN 2015, no. 19, 9696{9726; MR3431607] (also see [Qu^oc-AnhNg^oand´ Van Hoang Nguyen, \Sharp reversed Hardy- n Littlewood-Sobolev inequality on R ", preprint, arXiv:1508.02041] for a different proof), it is not hard to see that the functional Jα,R is bounded from below in the set of positive, integrable, and antipodally symmetric functions f. Then by a density argument and variational techniques, the infimum of Jα,R is achieved by a positive and antipodally symmetric function. In the next part of the paper, to pave the way for further research, the author carefully derives several properties, such as the conformal covariant property of the corresponding integral operator of the form Z α−n cn,α jξ − ηj u(η)dσg n S −1 R α−n with c = n jξ − ηj dσg n , for any metric g conformally covariant to the standard n,α S S n n−2 n metric gS on S . Knowing that cn;2jx − yj is the Green function for the Laplacian n ∆ on R with n > 3 and that the Green function Gg of the conformal Laplacian −∆g + n−2 4(n−1) scalg enjoys the conformally covariant property Ggb(y; x) = φ−1(y)φ−1(x)Gg(y; x) 4=(n−2) if two metrics g and bg are conformal in the sense that bg = φ g, the integral operator Z g α−n [G (y; x)] 2−n u(y)dµg M n defined on any compact Riemannian manifold (M n; g) is also conformally covariant. In the last part of the paper, the author formally proposes the α-curvature Q α,gb problem, which asks whether there exists a function u on a given compact Riemannian n manifold (M ; g) of dimension n 6= 2 with positive scalar curvature scalg which solves Z g α−n n+α (3) u(x) = [G (y; x)] 2−n Q (y)u(y) n−α dµ α,gb g M n for a prescribed function Q . It is worth noting that (3) on Rn was studied earlier α,gb by Xu in his pioneering work, where several non-existence results were obtained [see J. Funct. Anal. 247 (2007), no. 1, 95{109; MR2319755]. Qu[U+1ED1]c Anh Ng^o References 1. J. Ai, K-S. Chou, and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var., 13 (2000), 311{337. MR1865001 2. A. Bahri and J. M. Coron, The calar curvature problem on the standard three- dimensional sphere, J. Funct. Anal., 255 (1991), 106{172. MR1087949 3. T.P. Branson, Q-curvature and spectral invariants. 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