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A Classification of Positive Solutions of Some Integral Systems

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Abstract

In this paper, we investigate the symmetry of some nonlinear integral systems with Riesz potentials. With the method of moving planes, we study the symmetry of positive solutions in two cases, on R n and on bounded domains. These results can be extended to integral equations with Bessel potentials.

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References

  1. Alexandroff A.D.: A characteristic property of the spheres. Ann. Math. Pura. Appl. 58, 303–354 (1962)

    Article  Google Scholar 

  2. Adams R.: Sobolev Spaces, Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975)

    Google Scholar 

  3. Chen D., Ma L.: Radial symmetry and monotonicity for an integral equation. J. Math. Anal. Appl. 342, 943–949 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen D., Ma L.: Radial symmetry and uniqueness for positive solutions of a Schröinger type system. Math. Comput. Model. 49, 379–385 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caffarelli L., Gidas B., Spruck J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen W., Li C.: Regularity of solutions for a system of integral equations. Commun. Pure Appl. Anal. 4, 1–8 (2005)

    MathSciNet  Google Scholar 

  7. Chen W., Li C., Ou B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59, 330–343 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen W., Li C., Ou B.: Classification of solutions for a system of integral equations. Commun. Partial Differ. Equ. 30, 59C65 (2005)

    Article  MathSciNet  Google Scholar 

  9. Chen W., Li C., Ou B.: Qualitative properties of solutions for an integral equation. Disc. Cont. Dyn. Syst. 12, 347–354 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jin C., Li C.: Symmetry of solutions to some systems of integral equations. Proc. Am. Math. Sci. 134, 1661–1670 (2005)

    Article  MathSciNet  Google Scholar 

  12. Li C., Lim J.: The singularity analysis of solutions to some integral equations. Commun. Pure Appl. Anal. 6, 453–464 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lin C.: A classification of solutions of a conformally invariant fourth order equation in R n. Comment. Math. Helv. 73, 206–231 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li D., Ströhmer G., Wang L.: Symmetry of integral equations on bounded domains. Proc. Am. Math. Soc. 137, 3695–3702 (2009)

    Article  MATH  Google Scholar 

  15. Li Y.: Prescribing scalar curvature on S n and related problems, part 1. J. Differ. Equ. 120, 319–410 (1995)

    Article  MATH  Google Scholar 

  16. Serrin J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Xiaotao Huang.

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This work was completed with the support NSFC: 10771166.

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Ma, F., Huang, X. & Wang, L. A Classification of Positive Solutions of Some Integral Systems. Integr. Equ. Oper. Theory 69, 393–404 (2011). https://doi.org/10.1007/s00020-010-1845-0

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  • DOI: https://doi.org/10.1007/s00020-010-1845-0

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