  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.
