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

TL;DR

This paper proves symmetry and monotonicity of positive solutions for a Lane-Emden system involving the logarithmic Laplacian, using a direct moving planes method and establishing key principles for this operator.

Contribution

It introduces a novel application of the moving planes method to the logarithmic Laplacian, demonstrating symmetry results for Lane-Emden systems with this operator.

Findings

Positive solutions are symmetric and monotone.

Key principles like maximum and narrow region principles are established.

Results extend to generalized Lane-Emden systems.

Abstract

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 as a 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 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.