Classification of solutions to a weighted singular fractional problem in the half space

Abstract

We focus on the classification of positive solutions to $(-\Delta {)}^{s}u=\frac{x_n^{\alpha }}{u^{\gamma }}$ in the half space with $\gamma >0$, subject to the Dirichlet condition. We show that when $-2s<\alpha <(\gamma -1)s$, all positive solutions exhibit one-dimensional symmetry and are monotone increasing in $x_n$. Moreover, we provide a complete classification of all such one-dimensional solutions via their ``asymptotic $s$-order slope". When $\alpha$ lies outside this range, we demonstrate the nonexistence of global positive solutions.