关于RCD*(-K,N)空间上Lane-Emden方程组非常数解的存在性
On the existence of nonconstant solutions of system of Lane-Emden equations on $\mathrm{RCD}^*(-K,N)$ spaces
浏览论文内容
中文总结 AI 辅助
本文在RCD*(-K,N)空间上建立Lane-Emden方程组正解的椭圆梯度估计，证明刘维尔型定理，推广光滑黎曼空间的对应结果，研究该类空间上非线性椭圆方程组的解。
中文摘要 AI 辅助
本文针对满足合成里奇曲率维数条件RCD*(-K,N)的度量测度空间上Lane-Emden方程组的正解，建立了椭圆梯度估计。我们的方法结合了RCD*(-K,N)框架下的弱微分演算与博赫纳不等式，以及合适的辅助函数论证，将经典梯度估计技术推广到非光滑空间。作为应用，我们在适当的几何假设下证明了正解的刘维尔型定理，该结果有助于确定常数解存在的约束条件。我们还提及后续可能存在非常数解的若干情形。这些结果推广了光滑黎曼框架下的对应结论，为具有合成里奇曲率下界空间上的非线性椭圆方程组研究作出贡献。
英文摘要
In this article, we establish elliptic gradient estimates for positive solutions of the Lane-Emden system on metric measure spaces satisfying the synthetic Ricci curvature-dimension condition $\mathrm{RCD}^*(-K,N)$. Our approach combines the weak differential calculus and the Bochner inequality available in the $\mathrm{RCD}^*(-K,N)$ setting with suitable auxiliary function arguments, extending classical gradient estimate techniques to nonsmooth spaces. As an application, we prove a Liouville-type theorem for positive solutions under appropriate geometric assumptions. This result helps us to identify constraints for which constant solutions exist. We also mention some cases where nonconstant solutions may exist in sequel. These results generalize corresponding results from the smooth Riemannian setting and contribute to the study of nonlinear elliptic systems on spaces with synthetic Ricci curvature lower bounds.
