zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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).

MSC:
45G05Singular nonlinear integral equations
45M20Positive solutions of integral equations
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
WorldCat.org
Full Text: DOI
References:
[1] Ai, J.; Chou, K.; Wei, J.: Self-similar solutions for the anisotropic affine curve shortening problem. Calc. var. Partial differential equations 13, 311-337 (2001) · Zbl 1086.35035
[2] Branson, T. P.: Group representations arising from Lorentz conformal geometry. J. funct. Anal. 74, 199-291 (1987) · Zbl 0643.58036
[3] Branson, T.; Chang, S. -Y.A.; Yang, P. C.: Estimates and extremal problems for the log-determinant on 4-manifolds. Comm. math. Phys. 149, 241-262 (1992) · Zbl 0761.58053
[4] Brezis, H.; Merle, F.: Uniform estimates and below-up behavior for solutions of -${\Delta}u=v(x)$eu in two dimensions. Comm. partial differential equations 16, 1223-1253 (1991) · Zbl 0746.35006
[5] Caffarelli, L.; Gidas, B.; Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. pure appl. Math. 42, 271-297 (1989) · Zbl 0702.35085
[6] Chang, S. -Y.A.; Yang, P. C.: Extremal metrics of zeta functional determinants on 4-manifolds. Ann. math. 142, 1012-1171 (1995)
[7] Chang, S. -Y.A.; Yang, P. C.: On uniqueness of solution of a n-th order differential equation in conformal geometry. Math. res. Lett. 4, 91-102 (1997) · Zbl 0903.53027
[8] Chen, W.; Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke math. J. 63, 615-622 (1991) · Zbl 0768.35025
[9] Y.S. Choi, X. Xu, Nonlinear biharmonic equation with negative exponent, 1999, preprint.
[10] C. Fefferman, Graham, Conformal invariants, in: Élie Cartan et les Mathématiques d’aujourd’hui, Asterisque (1985) 95-116.
[11] Gidas, B.; Ni, N. W.; Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. math. Phys. 68, 209-243 (1979) · Zbl 0425.35020
[12] Gidas, B.; Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. pure appl. Math. 34, 525-598 (1981) · Zbl 0465.35003
[13] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Grundlehren der Mathematischen Wissen Schaften, vol. 224, Springer, Berlin, 1983. · Zbl 0562.35001
[14] Gursky, M.: The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics. Ann. math. 148, 315-337 (1998) · Zbl 0949.53025
[15] Lin, C. S.: A classification of solutions of a conformally invariant fourth order equation in rn. Comment. math. Helv. 73, 206-231 (1998) · Zbl 0933.35057
[16] Li, Y.; Zhu, M.: Uniqueness theorems through the method of moving spheres. Duke math. J. 80, 383-417 (1995) · Zbl 0846.35050
[17] P.J. McKenna, W. Reithel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differential Equations (37) (2003) 13pp (electronic). · Zbl 1109.35321
[18] Ou, B.: Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition. Differential integral equations 9, 1157-1164 (1996) · Zbl 0853.35045
[19] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, 1983, preprint. · Zbl 1145.53053
[20] Pucci, P.; Serrin, J.: Critical exponents and critical dimensions for polyharmonic operators. J. math. Pures appl. 69, 55-83 (1990) · Zbl 0717.35032
[21] Stein, E. M.: Singular integrals and differentiability properties of functions. (1970) · Zbl 0207.13501
[22] Wei, J.; Xu, X.: Classification of solutions of higher order conformally invariant equation. Math. ann. 313, 207-228 (1999) · Zbl 0940.35082
[23] Xu, X.: Uniqueness theorem for the entire positive solutions of biharmonic equations in rn. Roy. soc. Edin. proc. A (Mathematics) 130A, 651-670 (2000) · Zbl 0961.35037
[24] X. Xu, Classification of solutions of certain fourth order nonlinear elliptic equations in R4, Pacific J. Math., to appear.
[25] Xu, X.; Yang, P.: On a fourth order equation in 3-D. ESAIM control optim. Calc. var. 8, 1029-1042 (2002) · Zbl 1071.53526