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

Rong Zhang; Vishvesh Kumar; Michael Ruzhansky

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arXiv · 2210.09110

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

Rong Zhang·Vishvesh Kumar·Michael Ruzhansky

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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.

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Rong Zhang, Vishvesh Kumar, Michael Ruzhansky. 2022-10-17. Symmetry of positive solutions for Lane-Emden systems involving the Logarithmic Laplacian. https://arxiv.org/abs/2210.09110

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