Non-degeneracy of multi-bump solutions for the prescribed scalar curvature equations and applications

TL;DR

This paper investigates the non-degeneracy of multi-bump solutions to prescribed scalar curvature equations in Euclidean space, providing new methods and improved results that have applications in geometric analysis.

Contribution

It introduces modified methods to establish non-degeneracy of multi-bump solutions, enhancing previous results and broadening their applicability.

Findings

Proved non-degeneracy of multi-bump solutions under new conditions

Developed improved analytical techniques for scalar curvature equations

Extended applications to geometric problems involving prescribed curvature

Abstract

We consider the following prescribed scalar curvature equations in ${\mathbb{R}}^N$ $$ - \Delta u =K(|y|)u^{2^*-1},\quad u>0 \quad \mbox{in} \quad {\mathbb{R}}^N, \quad u \in D^{1, 2}({\mathbb{R}}^N), $$ where $K(r)$ is a positive function, $2^*=\frac{2N}{N-2}$. We modify the methods and improve the results in [15].