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On the Sixth-Order Joseph–Lundgren Exponent

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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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Acknowledgments

The authors would like to thank Professor Dong Ye to send them the preprint of Professor Simon Blatt which is a basic reference in this paper. They are also grateful to Professors Ali Maalaoui and Louis Dupaigne for improving the English of this paper.

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Authors and Affiliations

  1. Institut Supérieur des Mathématiques Appliquées et de l’Informatique, Université de Kairouan, Kairouan, Tunisia

    Abdellaziz Harrabi

  2. Faculté des Sciences, Département de Mathématiques, Université de Sfax, B.P 1171, 3000, Sfax, Tunisia

    Belgacem Rahal

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  1. Abdellaziz Harrabi
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Correspondence to Abdellaziz Harrabi.

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Communicated by Nader Masmoudi.

In Memory of Our Great Professor Abbas Bahri, with Gratitude.

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Harrabi, A., Rahal, B. On the Sixth-Order Joseph–Lundgren Exponent. Ann. Henri Poincaré 18, 1055–1094 (2017). https://doi.org/10.1007/s00023-016-0522-5

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