Heat flow method to Lichnerowicz type equation on closed manifolds

Abstract

In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -Δu=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1,q>0$, and $A(x)>0$, $B(x)\ge 0$ are given smooth functions. Our analysis is based on the global existence of positive solutions to the following heat equation {ll} u_t-Δu=A(x)u^{-p}-B(x)u^{q},\quad in\quad M\times\mathbb{R}^{+}, u(x,0)=u_0,\quad in\quad M with the positive smooth initial data $u_0$.

10 pages

References in corpus (2)

• A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds
• The constraint equations for the Einstein-scalar field system on compact manifolds