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Liouville type theorem for higher-order elliptic system with Navier boundary condition

  • Weiwei Zhao
  • Jinge Yang
  • Sining Zheng
Article
  • 223 Downloads

Abstract

This paper deals with the Liouville theorem for a higher-order elliptic system in the half-space subject to the Navier boundary value conditions. We obtain this via establishing the Liouville type theorem for the equivalent integral system by the moving plane method.

Keywords

Higher-order elliptic system Liouville type theorem Navier boundary value Moving planes method in integral form Rotational symmetry 

Mathematics Subject Classification

Primary 35J58 Secondary 35J61 

References

  1. 1.
    Berestycki H., Capuzzo Dolcetta I., Nirenberg L.: Problèmes elliptiques indéfinis et théorèmes de Liouville non linèaires. C. R. Acad. Sci. Paris Série. I Math. 317, 945–950 (1993)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Berestycki H., Capuzzo Dolcetta I., Nirenberg L.: Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal. 4, 59–78 (1994)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Birindelli I., Mitidieri E.: Liouville theorems for elliptic inequalities and applications. Proc. Roy. Soc. Edinburgh 128, 1217–1247 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brason T.: Differential operators canonically associated to a conformal structure. Math. Scand. 2, 293–345 (1985)Google Scholar
  5. 5.
    Cao L., Chen W.: Liouville type theorems for poly-harmonic Navier problems. Discrete Contin. Dyn. Syst. 33, 3937–3955 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cao L., Dai Z.: A Liouville-type theorem for an integral equation on a half-space \({\mathbb{R}^{n}_{+}}\) . J. Math. Anal. Appl. 389, 1365–1373 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Caristi G., D’Ambrosio L., Mitidieri E.: Representation formula for solutions to some classes of higher order systems and related Liouville theorems. Milan J. Math. 76, 27–67 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chen W., Li C., Ou B.: Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59, 330–343 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Chen W., Fang Y., Li C.: Super poly-harmonic property of solutions for Navier boundary problems on a half space. J. Funct. Anal. 265, 1522–1555 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chen Z., Zhao Z.: Potential theory for elliptic systems. Ann. Probab. 24, 293–319 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    de Figueiredo D., Felmer P.: A Liouville-type theorem for elliptic systems. Ann. Scuola Norm. Sup. Pisa CI. Sci. 21, 387–397 (1994)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Djadli Z., Malchiodi A., Almedou M.: Prescribing a fourth order conformal invariant on the standard sphere. II. Blow up analysis and applications. Ann. Sc. Norm. Super. Pisa CI. Sci. 1, 387–434 (2002)zbMATHGoogle Scholar
  13. 13.
    Domingos A.R., Guo Y.: A note on a Liouville type result for a system of fourth-order equations in \({\mathbb{R}^N}\) . Electron. J. Differ. Equ. 99, 1–20 (2002)MathSciNetGoogle Scholar
  14. 14.
    Esposito P., Robert F.: Mountain pass critical points for Paneitz-Branson operators. Calv. Var. Partial Differ. Equ. 15, 493–517 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Fang Y., Chen W.: A Liouville type theorem for poly-harmonic Dirichlet problems in a half space. Adv. Math. 229, 2835–2867 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Gidas B., Spruck J.: A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differ. Equ. 6, 525–598 (1981)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Jin C., Li C.: Symmetry of solutions to some integral equations. Proc. Am. Math. Soc. 134, 1661–1670 (2005)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Lenhart S., Belbas S.: A system of nonlinear partial differential equations arising in the optimal control of stochastic systems. SIAM J. Appl. Math. 43, 465–475 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Li, C., Ma, L.: Uniqueness of positive bound states to shrödinger systems with critical exponents. SIAM J. Math. Anal. 40:1049–1057 (2008)Google Scholar
  20. 20.
    Liu J., Guo Y., Zhang Y.: Liouville-type theorem for polyharmonic systems in \({\mathbb{R}^N}\) . J. Differ. Equ. 225, 685–709 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Lu G., Wang P., Zhu J.: Liouville-type theorems and decay estimates for solutions to higher order elliptic equations. Ann. I. H. Poincaré-AN 29, 653–665 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Ma L., Chen D.: A Liouville type theorem for an integral system. Commun. Pure Appl. Anal. 5, 855–859 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Mitidieri E.: A Rellich type identity and applications. Comm. Partial Differ. Equ. 18, 125–151 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Mitidieri E.: Nonexistence of positive solutions of semilinear elliptic systems in \({\mathbb{R}^{n}}\) . Differ. Integral Equ. 9, 465–479 (1996)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Nazarov S., Sweers G.: A hinged plate equation and iterated Dirichlet Laplace operator on domanins with concave corners. J. Differ. Equ. 233, 151–180 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Peletier, L.: Nonlinear eigenvalue problems for higher-order model equations. Handbook of Differential Equations, Stationary Partial Differential Equations, Volume 3, Chapter 7, Ed. M. Chipot and P. Quittner, Elsevier (2006)Google Scholar
  27. 27.
    Peter P., Quittner P., Souplet Ph.: Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: elliptic equations and systems. Duke Math. J. 139, 555–579 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Peletier, L.; Troy, W.; Patterns, S.: Higher order models in physics and mechanics, Progress in Nonlinear Differential Equations and their Applications. Birkhauser Boston, Inc., Boston, MA, 45 (2001)Google Scholar
  29. 29.
    Reichel W., Weth T.: A prior bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems. Math. Z. 261, 805–827 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Reichel W., Weth T.: Existence of solutions to nonlinear, subcritical higher-order elliptic dirichlet problems. J. Differ. Equ. 248, 1866–1878 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Serrin J., Zou H.: Nonexistence of positive solutions of Lane-Emden systems. Differ. Integral Equ. 9, 635–653 (1996)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Sirakov B.: Existence results and a priori bounds for higher order elliptic equations and systems. J. Math. Pures Appl. 89, 114–133 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Berg J.B.: The phase-plane picture for a class of fourth-order conservative differential equations. J. Differ. Equ. 161, 110–153 (2000)CrossRefzbMATHGoogle Scholar
  34. 34.
    Wei J., Xu X.: Classification of solutions of higher order conformally invariant equations. Math. Ann. 313, 207–228 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Zhang Y.: A Liouville type theorems for poly-harmonic elliptic systems. Electron. J. Differ. Equ. 99, 1–20 (2002)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianPeople’s Republic of China
  2. 2.Department of ScienceNanchang Institute of TechnologyNanchangPeople’s Republic of China

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