Symmetry of Positive Solutions for Lane-Emden Systems Involving the Logarithmic Laplacian

Abstract

We study the Lane-Emden system involving the logarithmic Laplacian:

$$ \textstyle\begin{cases} \ \mathcal{L}_{\Delta }u(x)=v^{p}(x) ,& x\in \mathbb{R}^{n}, \\ \ \mathcal{L}_{\Delta }v(x)=u^{q}(x) ,& x\in \mathbb{R}^{n}, \end{cases} $$

where \(p,q>1\), \(n\geq 2\) and \(\mathcal{L}_{\Delta }\) denotes the logarithmic Laplacian arising as a formal derivative \(\partial _{s}|_{s=0}(-\Delta )^{s}\) of the fractional Laplacian \((-\Delta )^{s}\) 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.