Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs - Duong, Anh Tuan; Fujiié, Setsuro (Comptes Rendus. Mathématique)
Let $(V,E)$ be a locally finite weighted graph. We study some qualitative properties of positive solutions of the Lichnerowicz equation \[ v_t-\Delta v=v^{-p-2}-v^p, \;(x,t)\in V \times \mathbb{R}, \] and of (sign-changing) solutions of the Ginzburg-Landau system \[ {\left\lbrace \begin{array}{ll} u_t-\Delta u=u-u^3-\lambda uv^2 , \;(x,t)\in V \times \mathbb{R},\\ v_t-\Delta v=v-v^3-\lambda vu^2, \;(x,t)\in V \times \mathbb{R}, \end{array}\right.} \] where $p>0$, $\lambda >0$ and $\Delta $ is the standard discrete graph Laplacian. Firstly, we prove that any positive solution $v$ of the Lichnerowicz equation satisfies $v\ge 1$. Moreover, if we assume the boundedness of positive solution $v$, then it must be trivial, i.e $v\equiv 1$. We also construct a nontrivial positive solution of the Lichnerowicz equation to show that the boundedness assumption is necessary. Secondly, we obtain sharp upper bound for solutions of the Ginzburg-Landau system depending on the range of $\lambda $.