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--Szegö 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.

Keywords

  1. ground states
  2. symmetry
  3. uniqueness

MSC codes

  1. 35Q55

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 6119 - 6135
ISSN (online): 1095-7154

History

Submitted: 30 March 2022
Accepted: 20 July 2022
Published online: 17 November 2022

Keywords

  1. ground states
  2. symmetry
  3. uniqueness

MSC codes

  1. 35Q55

Authors

Affiliations

Funding Information

Swiss National Science Foundation : 20021-169464

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