Uniqueness results for higher order Lane-Emden systems

In this paper we develop a Gidas–Ni–Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace op...

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Bibliographic Details
Published in:Nonlinear analysis 2020-09, Vol.198, p.111871, Article 111871
Main Authors: Cassani, Daniele, Schiera, Delia
Format: Article
Language:English
Subjects:
Boundary conditions
Dirichlet problem
Gidas–Ni–Nirenberg type results
Higher order elliptic operators
Mathematical analysis
Uniqueness
Variational and non-variational elliptic systems
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Summary:In this paper we develop a Gidas–Ni–Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet’s: the obstruction is apparently a new phenomenon and seems due to some loss of information. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier’s boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2020.111871