Heat flow method to Lichnerowicz type equation on closed manifolds

TL;DR

This paper proves the existence of positive solutions to a nonlinear Lichnerowicz-type equation on closed manifolds by analyzing a related heat equation and establishing global solutions.

Contribution

It introduces a novel approach using heat equation methods to establish existence results for a class of nonlinear elliptic equations on closed manifolds.

Findings

Existence of positive solutions under certain conditions

Global solutions to the associated heat equation are established

Method can be applied to similar nonlinear equations

Abstract

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 $u_0$.