Multiple solutions to polyharmonic elliptic problem involving GJMS operator on compact manifolds


Using variational methods, we prove existence and multiplicity of solutions to polyharmonic elliptic problem involving GJMS operator on smooth compact Riemannian manifold. An application is given in the Euclidean context.

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  1. 1.

    Ambrosetti, A.: Critical points and nonlinear variational problems. Soc. Mathem. de France, mémoire, 49, vol. 20, fascicule 2 (1992)

  2. 2.

    Ambrosetti, A., Azorero, J.G.: Multiplicity results for nonlinear elliptic equations.J. Funct. Anal. 137, 219–242 (1996)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Benalili, M., Zouaoui, A.: Elliptic equation with critical and negative exponents involving the GJMS operator on compact Riemannian manifolds. J Geom. Phy. 140, 56–73 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Branson, T.P.: The Functional Determinant, Lecture Notes Series, vol. 4. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1993)

    Google Scholar 

  5. 5.

    Clapp, M., Squassina, M.: Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data. Commun. Pure Appl. Anal. 2, 171–186 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ge, Y., Wei, J., Zhou, F.: A critical elliptic problem for polyharmonic operators. J. Funct. Anal. 260(8), 2247–2282 (2011)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Lions, P.L.: The concentration-compactness principle in the calculus of variations: the limit case. Part I. Rev. Mat. Iberoam. 1, 145–201 (1985)

    Article  Google Scholar 

  8. 8.

    Mazumdar, S.: GJMS-type operators on a compact Riemannian manifold: best constants and Coron-type solutions. J. Diff. Eq. 261(9) (2015)

  9. 9.

    Paneitz, S.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary). SIGMA 4(3), 036 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Robert, F.: Admissible \(Q\)-curvatures under Isometries for the Conformal GJMS Operators. Nonlinear Elliptic Partial Differential Equations. Contemp Math, vol. 540, pp. 241–259. American Mathematical Society, Providence (2011)

    Google Scholar 

  11. 11.

    Serrin, J., Pucci, P.: Critical exponents and critical dimensions for polyharmonic operators. J. Math. Pures Appl. (9) 69(1), 55–83 (1990)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Swanson, A.: The best Sobolev constant. Appl. Anal. 47(4), 227–239 (1992)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Tahri, K.: Nohomogenous polyharmonic elliptic problem involving GJMS operator on compact manifolds. Asian-Eur. J. Math. 13(1), 2050115 (2020)

    Google Scholar 

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Correspondence to Kamel Tahri.

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Tahri, K. Multiple solutions to polyharmonic elliptic problem involving GJMS operator on compact manifolds. Afr. Mat. 31, 437–454 (2020).

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  • Polyharmonic elliptic problems
  • GJMS operator
  • Riemannian manifold
  • multiple solutions