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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 $\mathbb R^n$. 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.

45E10Integral equations of the convolution type
35J65Nonlinear boundary value problems for linear elliptic equations
45M20Positive solutions of integral equations
35C15Integral representations of solutions of PDE
Full Text: DOI arXiv
[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 · 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) · Zbl 0527.42011
[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) · Zbl 0232.42007
[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