A Quantitative Gidas--Ni--Nirenberg Theorem for the Fractional Laplacian

We establish a quantitative Gidas--Ni--Nirenberg theorem for the pure fractional Dirichlet problem \[ (-Δ)^s u=k(x)g(u)\quad\text{in }B_1, \qquad u=0\quad\text{in }\mathbb R^n\setminus B_1, \qquad 0<s<1. \] For positive bounded solutions under a two-sided $L^\infty$ normalization, we prove \[ \mathc