# On symmetry and uniqueness of ground states for linear and nonlinear elliptic PDEs

## TL;DR

This paper establishes symmetry and uniqueness results for ground states of linear and nonlinear elliptic PDEs with arbitrary order, employing Fourier analysis and a recent Hardy-Littlewood majorant rigidity result, extending beyond traditional methods.

## Contribution

It introduces a novel approach using Fourier transform and a recent rigidity theorem to analyze ground states for higher order elliptic PDEs, surpassing classical techniques.

## Key findings

- Proved symmetry of ground states in nonlinear elliptic PDEs.
- Established uniqueness of ground states in linear elliptic PDEs.
- Extended analysis to PDEs of arbitrary order, including higher order cases.

## Abstract

We study ground state solutions for linear and nonlinear elliptic PDEs in $\mathbb{R}^n$ with (pseudo-)differential operators of arbitrary order. We prove a general symmetry result in the nonlinear case as well as a uniqueness result for ground states in the linear case. In particular, we can deal with problems (e.\,g. higher order PDEs) that cannot be tackled by usual methods such as maximum principles, moving planes, or Polya--Szeg\"o inequalities. Instead, we use arguments based on the Fourier transform and we apply a rigidity result for the Hardy-Littlewood majorant problem in $\mathbb{R}^n$ recently obtained by the last two authors of the present paper.