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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 5, pp 1675–1692 | Cite as

High-order Bahri–Lions Liouville-type theorems

  • Abdellaziz HarrabiEmail author
Article
  • 49 Downloads

Abstract

Consider the following polyharmonic equations:
$$\begin{aligned} (-\Delta )^r u=f(u)\ \ \ \text{ in }\,\, \mathcal {O},\qquad \qquad (0.1) \end{aligned}$$
where \(\mathcal {O}=\mathbb {R}^n\) or \(\mathcal {O}=\mathbb {R}_+^n\) with the Dirichlet boundary conditions, \(r \in \mathbb {N}^*\) and \(n\ge 2r+1\). We prove some Liouville-type theorems classifying stable (or stable at infinity) solutions, possibly unbounded and sign-changing. Regarding the class of stable solutions, we focus on the case of superlinear nonlinearities f with subcritical or critical growth near zero, like
$$\begin{aligned}&f(s)=|s|^{p-1}s(1+c_0|s|^{q}) \text{ or } f(s)=|s|^{p-1}s\exp (s^2),\\&\quad \text{ where } 1<p\le \frac{n+2r}{n-2r},\quad q>0 \text{ and } c_0\ge 0. \end{aligned}$$
Our approach to get the main integral estimates makes use of delicate analysis with appropriate test functions and weighted seminorms. We also establish a variant of Pohozaev identity (Pohozaev in Sov Math Dokl 5:1408–1411, 1965). This permits us to get classification result for stable at infinity solutions under the global subcritical condition:
$$\begin{aligned} \frac{2n}{n-2r}F(s) -f(s)s>0,\; \forall s\ne 0, \text{ where } F(s)=\int _{ 0}^{s}f(t)\mathrm{d}t. \end{aligned}$$
Our assumptions can be verified by many nonlinearities very close to the critical growth, like
$$\begin{aligned} f(s)= c|s|^{p-1}s+\dfrac{|s|^{\frac{4r}{n-2r}}s}{\ln ^q(s^2+a)}, \end{aligned}$$
where \(1<p<\frac{n+2r}{n-2r}, \;q \ge 0,\;c\ge 0 \text{ and } a>1,\) with \(\frac{4r}{n-2r}-\frac{2q}{a\ln (a)}>0; \text{ or } c>0 \text{ if } q=0\).

Keywords

Polyharmonic equations Liouville theorems Morse index Pohozaev identity Blow-up argument Universal estimate \(L^\infty \)-estimates 

Mathematics Subject Classification

Primary 35G20 35G30 Secondary 35B05 35B09 35B53 

Notes

Acknowledgements

This work was partially done in the I.C.T.P, in July 2017. The author wishes to acknowledge the reviewer for his/her thorough review, which significantly contributed to improving the quality of the publication.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentNorthern Borders UniversityArarSaudi Arabia
  2. 2.Institut Supérieur des Mathématiques Appliquées et de l’InformatiqueUniversité de KairouanKairouanTunisia
  3. 3.Abdus Salam International Centre for Theoretical PhysicsTriesteItaly

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