On the existence of nonconstant solutions of system of Lane-Emden equations on $\mathrm{RCD}^*(-K,N)$ spaces
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.