Annales Henri Poincaré

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

On the Sixth-Order Joseph–Lundgren Exponent

Article

Abstract

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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© 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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