| Abstract: |
| We focus on the classification of positive solutions to $(-\Delta)^s u=\frac{1}{u^\gamma}$ in the half space with $\gamma>0$, subject to the Dirichlet condition. We show that when $\gamma>1$, 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. |
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