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)
Radial symmetry and monotonicity for an integral equation. (English) Zbl 1140.45004
Authors’ summary: We study radial symmetry and monotonicity of positive solutions of an integral equation arising from some higher-order semilinear elliptic equations in the whole space ${ℝ}^{n}$. Instead of the usual method of moving planes, we use a new Hardy-Littlewood-Sobolev type inequality for the Bessel potentials to establish the radial symmetry and monotonicity results.
MSC:
 45E10 Integral equations of the convolution type 35J65 Nonlinear boundary value problems for linear elliptic equations 45M20 Positive solutions of integral equations 35C15 Integral representations of solutions of PDE
References:
 [1] Adams, R.: Sobolev spaces, Pure appl. Math. 65 (1975) · Zbl 0314.46030 [2] Bourgain, J.: Global solutions of nonlinear Schrödinger equations, Amer. math. Soc. colloq. Publ. 46 (1999) · Zbl 0933.35178 [3] 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 · doi:10.1002/cpa.3160420304 [4] Chen, W.; Li, C.; Ou, B.: Classification of solutions for a system of integral equations, Comm. partial differential equations 30, 59-65 (2005) · Zbl 1073.45005 · doi:10.1081/PDE-200044445 [5] Chen, W.; Li, C.; Ou, B.: Classification of solutions for an integral equation, Comm. pure appl. Math. 59, 330-343 (2006) · Zbl 1093.45001 · doi:10.1002/cpa.20116 [6] Du, Y.; Ma, L.: Some remarks related to de giorgi’s conjecture, Proc. amer. Math. soc. 131, 2415-2422 (2003) · Zbl 1094.35047 · doi:10.1090/S0002-9939-02-06867-3 [7] Gidas, B.; Ni, W.; Nirenberg, L.: Symmetry and related properties via the maximum principle, Comm. math. Phys. 68, 209-243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125 [8] Gidas, B.; Ni, W.; Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in rn, Adv. math. Suppl. stud. 7a, 369-402 (1981) · Zbl 0469.35052 [9] Kwong, M.: Uniqueness of positive solutions of ${\Delta }u$ - u+up=0 in rn, Arch. ration. Mech. anal. 105, 243-266 (1989) · Zbl 0676.35032 · doi:10.1007/BF00251502 [10] Li, Y.: Remark on some conformally invariant integral equations: the method of moving spheres, J. eur. Math. soc. (JEMS) 6, 153-180 (2004) · Zbl 1075.45006 · doi:10.4171/JEMS/6 · doi:http://www.ems-ph.org/journals/show_issue.php?issn=1435-9855&vol=6&iss=2 [11] Lieb, E.: Sharp constants in the Hardy – Littlewood – Sobolev and related inequalities, Ann. of math. 118, 349-374 (1983) [12] Ma, L.; Chen, D.: A Liouville type theorem for an integral system, Commun. pure appl. Anal. 5, 855-859 (2006) · Zbl 1134.45007 · doi:10.3934/cpaa.2006.5.855 [13] Ni, W.; Takagi, I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke math. J. 70, 247-281 (1993) · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4 [14] Stein, E.: Singular integrals and differentiability properties of functions, Princeton ser. Appl. math. 30 (1970) · Zbl 0207.13501 [15] Stein, E.; Weiss, G.: Introduction to Fourier analysis on Euclidean spaces, Princeton ser. Appl. math. 32 (1971) [16] Wei, J.; Xu, X.: Classification of solutions of higher order conformally invariant equations, Math. ann. 313, 207-228 (1999) · Zbl 0940.35082 · doi:10.1007/s002080050258 [17] Ziemer, W.: Weakly differentiable functions: Sobolev spaces and functions of bounded variation, Geom. topol. Monogr. 120 (1989) · Zbl 0692.46022