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

We establish a quantitative Gidas–Ni–Nirenberg theorem for the pure fractional Dirichlet problem $$(-\mathrm{\Delta }{)}^{s}u=k(x)g(u)\text{in }{B}_1,u=0\text{in }{\mathbb{R}}^n\setminus B_1,0<s<1.$$ For positive bounded solutions under a two-sided $L^{\mathrm{\infty }}$ normalization, we prove $$\mathcal{D}(u)\le C\mathcal{D}(k{)}^{\gamma },0<\gamma <1,$$ 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.