Morse index and non-degeneracy of double-tower solutions for prescribed scalar curvature problem

Abstract

We consider the following prescribed scalar curvature equations in $R^N$:(0.1)$-\Delta u=K\left(\right|y\left|\right)u^{2^{⁎}-1},u>0\text{ in }{R}^N,u\in D^{1,2}\left(R^N\right),$ where $K\left(r\right)$ is a positive function, $N\ge 5$ and $2^{⁎}=\frac{2N}{N-2}$. We are concerned with the solutions which are invariant under some non-trivial sub-group of $O\left(3\right)$ to the above problem (we call them double-tower solutions). We first prove a non-degeneracy result for the positive double-tower solutions. As an application, we consider an eigenvalue problem related to prescribed scalar curvature equations and investigate the properties of the eigenvalues. And we compute the Morse index of the double-tower solutions. Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of the Green function.