Symmetry of positive solutions for Lane-Emden systems involving the Logarithmic Laplacian

We study the Lane-Emden system involving the logarithmic Laplacian: $$\{\begin{array}{ll}{\mathcal{L}}_{\mathrm{\Delta }}u(x)=v^p(x),& x\in {\mathbb{R}}^n,\\ {\mathcal{L}}_{\mathrm{\Delta }}v(x)=u^q(x),& x\in {\mathbb{R}}^n,\end{array}$$ where $p,q>1$ and ${\mathcal{L}}_{\mathrm{\Delta }}$ denotes the Logarithmic Laplacian arising as a formal derivative ${\mathrm{\partial }}_s{\mathrm{\mid }}_{s=0}(-\mathrm{\Delta }{)}^s$ of fractional Laplacians at $s=0.$ By using a direct method of moving planes for the logarithmic Laplacian, we obtain the symmetry and monotonicity of the positive solutions to the Lane-Emden system. We also establish some key ingredients needed in order to apply the method of moving planes such as the maximum principle for anti-symmetric functions, the narrow region principle, and decay at infinity. Further, we discuss such results for a generalized system of the Lane-Emden type involving the logarithmic Laplacian.