Classification of anti-symmetric solutions to the fractional Lane-Emden system

We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian (−Δ)s (0 < s < 1). First we obtain a Liouville type theorem in the often-used defining space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal L}_{2s}}$$\end{document}. An interesting lower bound on the solutions is derived to estimate the Lipschitz coefficient in the sub-linear cases. Considering the anti-symmetric property, one can naturally extend the defining space from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal L}_{2s}}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal L}_{2s + 1}}$$\end{document}. Surprisingly, with this extension, we show the existence of non-trivial solutions. This is very different from the previous results of the Lane-Emden system.