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 $\mathrm{\Delta }u+f(u)=0$ 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 stability result for non-energy solutions to semilinear equations involving the Laplacian, even for the critical nonlinearity.