On Symmetry and Uniqueness of Ground States for Linear and Nonlinear Elliptic PDEs

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