# Uniqueness results for higher order elliptic equations and systems

## TL;DR

This paper extends the Gidas-Ni-Nirenberg technique to prove uniqueness of solutions for higher order polyharmonic equations and systems, revealing new phenomena for Dirichlet boundary conditions and establishing results for Navier conditions.

## Contribution

It develops a new method for proving uniqueness of solutions for polyharmonic equations up to eighth order and beyond, including systems, under various boundary conditions.

## Key findings

- Uniqueness proven for equations up to eighth order with Dirichlet boundary conditions.
- Extension of uniqueness results to arbitrary polyharmonic operators with natural boundary conditions.
- New existence results for systems of polyharmonic equations.

## Abstract

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 though far from being clear. 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.