A quantitative study of radial symmetry for solutions to semilinear equations in $\mathbb{R}^n$

A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations $Δu +f(u)=0$ in $\mathbb{R}^n$ must be radial and radially decreasing. In this paper, we consider both energy solutions in $\mathcal{D}^{1,2}(\mathbb{R}^n)$ and non-energy local weak solutions to small perturbations of these equations, and study its quantitative stability counterpart. To the best of our knowledge, the present work provides the first quantitative...