Strict Monotonicity of the First q-Eigenvalue of the Fractional p-Laplace Operator Over Annuli

Let $$B, B'\subset \mathbb {R}^d$$ with $$d\ge 2$$ be two balls such that $$B'\subset \subset B$$ and the position of $$B'$$ is varied within B. For $$p\in (1, \infty ),$$ $$s\in (0,1)$$ , and $$q \in [1, p^*_s)$$ with $$p^*_s=\frac{dp}{d-sp}$$ if $$sp < d$$ and $$p^*_s=\infty $$ if $$sp \ge d$$ , let $$\lambda ^s_{p,q}(B{\setminus } \overline{B'})$$ be the first q-eigenvalue of the fractional p-Laplace operator $$(-\Delta _p)^s$$ in $$B\setminus \overline{B'}$$ with the homogeneous nonlocal Dirichlet boundary conditions. We prove that $$\lambda ^s_{p,q}(B\setminus \overline{B'})$$ strictly decreases as the inner ball $$B'$$ moves towards the outer boundary $$\partial B$$ . To obtain this strict monotonicity, we establish a strict Faber-Krahn type inequality for $$\lambda _{p,q}^s(\cdot )$$ under polarization. This extends some monotonicity results obtained by Djitte-Fall-Weth (Calc. Var. Partial Differential Equations, 60:231, 2021) in the case of $$(-\Delta )^s$$ and $$q=1, 2$$ to $$(-\Delta _p)^s$$ and $$q\in [1, p^*_s).$$ Additionally, we provide the strict monotonicity results for the general domains that are difference of Steiner symmetric or foliated Schwarz symmetric sets in $$\mathbb {R}^d$$ .