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

TL;DR

This paper classifies positive solutions to a weighted fractional PDE in a half space, establishing symmetry, monotonicity, and nonexistence results based on parameter ranges.

Contribution

It provides a complete classification of solutions within certain parameter ranges and proves nonexistence outside those ranges.

Findings

All positive solutions are one-dimensional and monotone when parameters are in a specific range.

Complete classification of solutions via asymptotic slope.

Nonexistence of solutions outside the specified parameter range.

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.