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We study the Lane-Emden system involving the logarithmic Laplacian: $$\begin{cases} \ \mathcal{L}_{\Delta}u(x)=v^{p}(x) ,& x\in\mathbb{R}^{n},\\ \\mathcal{L}_{\Delta}v(x)=u^{q}(x) ,& x\in\mathbb{R}^{n}, \end{cases} $$ where$p,q>1$ and $\mathcal{L}_{\Delta}$ denotes the Logarithmic Laplacian arising asa formal derivative $\partial_s|_{s=0}(-\Delta)^s$ 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 theLane-Emden system. We also establish some key ingredients needed in order toapply the method of moving planes such as the maximum principle foranti-symmetric functions, the narrow region principle, and decay at infinity.Further, we discuss such results for a generalized system of the Lane-Emdentype involving the logarithmic Laplacian.

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