论文标题

涉及对数laplacian的车道填充系统的积极解决方案的对称性

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

论文作者

Zhang, Rong, Kumar, Vishvesh, Ruzhansky, Michael

论文摘要

我们研究了涉及对数laplacian的车道填充系统:$$ \ begin {case} \ \ m nathcal {l}_Δu(x)= v^{p}(x)(x),&x \ in \ mathbb {r} \ Mathcal {l}_ΔV(x)= u^{q}(x),&x \ in \ Mathbb {r}^{n} {n},\ end {cases} $ p,$ p,q>> 1 $ and $ \ nathcal {l} l} $ \ partial_s | _ {s = 0}( - δ)^s $ $ s = 0。我们还建立了一些关键成分,以应用移动平面的方法,例如反对称功能的最大原理,狭窄的区域原理和无穷大的衰减。此外,我们讨论了涉及对数laplacian的通用系统的广义系统。

We study the Lane-Emden system involving the logarithmic Laplacian: $$ \begin{cases} \ \mathcal{L}_Δu(x)=v^{p}(x) ,& x\in\mathbb{R}^{n},\\ \ \mathcal{L}_Δv(x)=u^{q}(x) ,& x\in\mathbb{R}^{n}, \end{cases} $$ where $p,q>1$ and $\mathcal{L}_Δ$ denotes the Logarithmic Laplacian arising as a formal derivative $\partial_s|_{s=0}(-Δ)^s$ 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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