The existence results for solutions of indefinite scalar curvature problem

TL;DR

This paper establishes new conditions under which solutions exist for the indefinite scalar curvature problem on Euclidean space, using bifurcation methods and Harnack inequalities to find positive solutions.

Contribution

It introduces novel conditions for the scalar curvature function ensuring the existence of solutions on $\mathbb{R}^n$, employing bifurcation and limit analysis techniques.

Findings

Existence of solutions under new conditions on the scalar curvature function.

Application of bifurcation method to obtain large solutions.

Use of Harnack inequality to analyze solutions near critical points.

Abstract

In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The key observation in our proof is that we use the bifurcation method to get a large solution and then after establishing the Harnack inequality for solutions near the critical points of the prescribed scalar curvature and taking limit, we find the nontrivial positive solution to the indefinite scalar curvature problem.