Fast decay solutons of Lane-Emden equations involving nonhomogeneous potentials

We are interested in studying the isolated singular solutions to Lane-Emden equation $$ -\Delta u= V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\}, $$ where $V$ is a nonhomogeneous potential satisfying some extra hypotheses. We proved that for any $\nu \in(0,+\infty)$, there exists $\nu$-fast decay solutions $u_{\nu}$. Furthermore, the limit function $u_{\infty}:=\lim_{\nu \to+\infty} u_{\nu}$ exists and is a slow decay solution.