Heat flow method to Lichnerowicz type equation on closed manifolds

Details Heat flow method to Lichnerowicz type equation on closed manifolds Heat flow method to Lichnerowicz type equation on closed manifolds

2010-02-27

arxiv.org/abs/1003.0053v1

Mathematics

Differential Geometry

10 pages

Scientific paper

In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -\Delta u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$, $B(x)\geq0$ are given smooth functions. Our analysis is based on the global existence of positive solutions to the following heat equation {ll} u_t-\Delta u=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 $u_0$.

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