Annales Henri Poincaré

, Volume 18, Issue 3, pp 1055–1094 | Cite as

On the Sixth-Order Joseph–Lundgren Exponent



In this paper, we study the solutions of the triharmonic Lane–Emden equation
$$\begin{aligned} -\Delta ^3 u=|u|^{p-1}u,\quad \text{ in }\;\; \mathbb {R}^n, \quad \text{ with }\;\;n\ge 2\quad \text{ and }\quad p>1. \end{aligned}$$
As in Dávila et al. (Adv. Math. 258:240–285, 2014) and Farina (J. Math. Pures Appl. 87:537–561, 2007), we prove various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of \(\mathbb {R}^n\). Again, following Dávila et al. (Adv. Math. 258:240–285, 2014), Hajlaoui et al. (On stable solutions of biharmonic prob- lem with polynomial growth. arXiv:1211.2223v2, 2012) and Wei and Ye (Math. Ann. 356:1599–1612, 2013), we first establish the standard integral estimates via stability property to derive the nonexistence results in the subcritical case by means of the Pohozaev identity. The supercritical case needs more involved analysis, motivated by the monotonicity formula established in Blatt (Monotonicity formulas for extrinsic triharmonic maps and the tri- harmonic Lane–Emden equation, 2014) (see also Luo et al., On the Triharmonic Lane–Emden Equation. arXiv:1607.04719, 2016), we then reduce the nonexistence of nontrivial entire solutions to that of nontrivial homogeneous solutions similarly to Dávila et al. (Adv. Math. 258:240–285, 2014). Through this approach, we give a complete classification of stable solutions and those which are stable outside a compact set of \(\mathbb {R}^n\) possibly unbounded and sign-changing. Inspired by Karageorgis (Nonlinearity 22:1653–1661, 2009), our analysis reveals a new critical exponent called the sixth-order Joseph–Lundgren exponent noted \(p_c(6,n)\). Lastly, we give the explicit expression of \(p_c(6,n)\). Our approach is less complicated and more transparent compared to Gazzola and Grunau (Math. Ann. 334:905–936, 2006) and Gazzola and Grunau (Polyharmonic boundary value problems. A monograph on positivity preserving and nonlinear higher order elliptic equations in bounded domains. Springer, New York, 2009) in terms of finding the explicit value of the fourth-Joseph–Lundgren exponent, \(p_c(4,n)\).


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  1. 1.
    Adimurthi, A., Santra, S.: Generalized Hardy–Rellich inequalities in critical dimension and its applications. Commun. Contemp. Math. 11(3), 367–394 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Angelsberg, G.: A monotonicity formula for stationary biharmonic maps. Math. Z. 252(2), 287–293 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arioli, G., Gazzola, F.: Grunau, HCh.: Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity. J. Differ. Equ. 230, 743–770 (2006)ADSCrossRefMATHGoogle Scholar
  4. 4.
    Bahri, A., Lions, P.L.: Solutions of superlinear elliptic equations and their Morse indices. Commun. Pure Appl. Math. 45, 1205–1215 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Blatt, S.: Monotonicity formulas for extrinsic triharmonic maps and the triharmonic Lane–Emden equation (2014) (submitted)Google Scholar
  6. 6.
    Dunham, W.: Cardano and the solution of the cubic equations. In: Journey through Genius: The Great Theorems of Mathematics (Chap. 6), pp. 133–154. Wiley, New York (1990). ISBN: 978-0-471-50030-8Google Scholar
  7. 7.
    Cowan, C., Ghoussoub, N.: Regularity of semi-stable solutions to fourth order nonlinear eigevalue problems on general domains. Calc. Var. doi:10.1007/s00526-012-0582-4
  8. 8.
    Cowan, C., Esposito, P., Ghoussoub, N.: Regularity of extremal solutions in fourth order nonlinear eigevalue problems on general domains. DCDS-A 28, 1033–1050 (2010)CrossRefMATHGoogle Scholar
  9. 9.
    Chang, A., Wang, L., Yang, P.C., Yung, S.: A regularity theory of biharmonic maps. Commun. Pure Appl. Math. 52(9), 1113–1137 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, W., Li, C.: Classification of solutions of some nonlinear nonlinear elliptic equations. Duke Math. J. 63, 615–622 (1991)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Damascelli, L., Gladiali, F.: Some nonexistence results for positive solutions of elliptic equations in unbounded domains. Rev. Mat. Iberoam. 20, 67–86 (2004)MathSciNetMATHGoogle Scholar
  12. 12.
    Dancer, E.N.: Superlinear problems on domains with holes of asymptotic shape and exterior problems. Math. Z. 229(3), 475–491 (1998)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dávila, J., Dupaigne, L., Farina, A.: Partial regularity of finite Morse index solutions to the Lane-Emden equation. J. Funct. Anal. 261, 218–232 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dávila, J., Dupaigne, L., Wang, K., Wei, J.: A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem. Adv. Math. 258, 240–285 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dupaigne, L., Ghergu, M., Goubet, O., Warnault, G.: Entire large solutions for semilinear elliptic equations. J. Differ. Equations 253, 2224–2251 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dupaigne, L., Harrabi, A.: The Lane–Emden Equation in Strips. Proc. R. Soc. Edin. Sec A (2016) (to appear in)Google Scholar
  17. 17.
    Evans, L.C.: Partial regularity for stationary harmonic maps into spheres. Arch. Ration. Mech. Anal. 116(2), 101–113 (1991)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Farina, A.: On the classification of solutions of the Lane-Emden equation on unbounded domains of \(\mathbb{R}^n\). J. Math. Pures Appl. 87, 537–561 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Farina, A., Ferrero, A.: Existence and stability propertiesof entire solutions to the polyharmonic equation \((-\Delta )^{m} u= e^u\) for any \(m\ge 1\). Ann. I. H. Poincaré (2014). doi:10.1016/j.anihpc.2014.11.005
  20. 20.
    Ferrero, A.: Grunau, HCh.: The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity. J. Differ. Equ. 234, 582–606 (2007)ADSCrossRefMATHGoogle Scholar
  21. 21.
    Ferrero, A., Grunau, HCh., Karageorgis, P.: Supercritical biharmonic equations with power-like nonlinearity: Ann. Mat. Pura Appl. 188, 171–185 (2009)Google Scholar
  22. 22.
    Gazzola, F., Grunau, HCh.: Radial entire solutions for supercritical biharmonic equations. Math. Ann. 334, 905–936 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Gazzola, F., Grunau, H. Ch., Sweers, G.: Polyharmonic boundary value problems. A monograph on positivity preserving and nonlinear higher order elliptic equations in bounded domains. Springer, New York (2009)Google Scholar
  24. 24.
    Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525–598 (1981)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Gidas, B., Ni, W., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Giga, Y., Kohn, R.V.: Asymptotically self-similar blow-up of semilinear heat equations. Commun. Pure Appl. Math. 38(3), 297–319 (1985)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Guo, Z., Wei, J.: Qualitative properties of entire radial solutions for a biharmonic equation with supcritical nonlinearity. Proc. Am. Math. Soc. 138, 3957–3964 (2010)CrossRefMATHGoogle Scholar
  28. 28.
    Hajlaoui, H., Harrabi, A., Ye, D.: On stable solutions of biharmonic problem with polynomial growth (2012). arXiv:1211.2223v2
  29. 29.
    Joseph, D.D., Lundgren, T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269 (1972/1973)Google Scholar
  30. 30.
    Karageorgis, P.: Stability and intersection properties of solutions to the nonlinear biharmonic equation. Nonlinearity 22, 1653–1661 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lin, C.S.: A classification of solutions to a conformally invariant equation in \(\mathbb{R}^4\). Commun. Math. Helv. 73, 206–231 (1998)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Luo, S., Wei, J., Zou, W.: On the Triharmonic Lane–Emden Equation (2016). arXiv:1607.04719
  33. 33.
    Pacard, F.: Partial regularity for weak solutions of a nonlinear elliptic equation. Manuscr. Math. 79(2), 161–172 (1993)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Polácik, P., Quittner, P., Souplet, P.: Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems. Duke Math. J. 139(3), 555–579 (2007)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Ramos, N., Rodrigues, P.: On a fourth order superlinear elliptic problem. Electr. J. Differ. Equations Conf. 06, 243–255 (2001)MathSciNetMATHGoogle Scholar
  36. 36.
    Rellich, F.: Perturbation theory of eigenvalue problems. Gordon and Breach, New York (1969) (MR 39 \(\sharp \) 2014 Zbl 0181.42002)Google Scholar
  37. 37.
    Souplet, P.: The proof of the Lane-Emden conjecture in four space dimensions. Adv. Math. 221, 1409–1427 (2009)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Wang, X.: On the Cauchy problem for reaction-diffusion equations. Trans. Am. Math. Soc. 337(2), 549–590 (1993)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Wei, J., Xu, X.: Classification of solutions of high order conformally invariant equations. Math. Ann. 313(2), 207–228 (1999)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Wei, J., Ye, D.: Liouville Theorems for stable solutions of biharmonic problem. Math. Ann. 356(4), 1599–1612 (2013)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Wei, J., Xu, X., and Yang, W.: On the classification of stable solutions to biharmonic problems in large dimensions. Pacific J. Math. 263(2), 495–512 (2013) (MR 3068555 Zbl 06196725)Google Scholar

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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Institut Supérieur des Mathématiques Appliquées et de l’InformatiqueUniversité de KairouanKairouanTunisia
  2. 2.Faculté des Sciences, Département de MathématiquesUniversité de SfaxSfaxTunisia

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