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

Abstract

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 \[ \mathcal D(u)\le C\,\mathcal D(k)^γ,\qquad 0<γ<1, \] where $\mathcal D$ measures spherical oscillation and outward radial increase. To the best of our knowledge, this provides the first quantitative Gidas--Ni--Nirenberg estimate for the pure fractional Dirichlet problem. The key new ingredient is a shift-uniform, weak Harnack inequality for shifted antisymmetric supersolutions, which overcomes the exterior-tail obstruction in the quantitative moving-plane argument. This mechanism extends to nonseparable nonlinearities.