Multiple high energy solutions of a nonlinear Hardy–Sobolev critical elliptic equation arising in astrophysics

In this article, we study the existence and multiplicity of high energy solutions to the problem proposed as a model for the dynamics of galaxies: $-\Delta u+V\left(x\right)u=\frac{{\left|u\right|}^{2_{\ast }-2}u}{\left|y\right|},x=(y,z)\in R^m\times R^{n-m},$where $n>4$, $2\le m<n$, $2_{\ast }≔\frac{2(n-1)}{n-2}$ and potential function $V\left(x\right):R^n\to R$. Benefiting from a global compactness result, we show that there exist at least two positive high energy solutions. Our proofs are based on barycenter function, quantitative deformation lemma and Brouwer degree theory.