In a seminal work published in 1930, the astrophysicist Takehiko Matukuma proposed the equation \(\begin{aligned} -\Delta u = \frac{1}{1+|x|^2} u^\alpha \quad \text {in } \textbf{R}^3 \end{aligned}\) as a mathematical model to describe the dynamics of globular cluster of stars. Here \(\alpha >1\) and \(u>0\) is the gravitational potential with \(\begin{aligned} \frac{1}{4\pi } \int _{\textbf{R}^3} \frac{1}{1+|x|^2}u^\alpha dx \end{aligned}\) representing the total mass. While the case \(\alpha >1\) has been at the center of the research of the equation and many interesting results have been drawn, much less is known in the regime \(\alpha \le 1\) . In this work, we are interested \(C^2\) -solutions to the equation in the case \(\alpha \le 1\) and \(n \ge 3\) . We prove that actually the equation originally posed in \(\textbf{R}^3\) admits no classical solution if \(\alpha \le 1\) . However, this is no longer true in higher-dimensional spaces \(n \ge 4\) . We also highlight further intriguing differences between the cases \(\textbf{R}^3\) and \(\textbf{R}^n\) with \(n \ge 4.\)
