In this work, we investigate the results regarding the existence and non-existence of non-negative, non-trivial, \(C^2\) - solutions to the equation \( -\Delta u = \bigg (\frac{1}{1+|x|^2}\bigg )^\sigma u^\alpha \quad \text {in } \textbf{R}^n \) with \(n \ge 3\) , \(\sigma \in (0,1)\) , and \(\alpha >1\) . Our choice of this mathematical model, called Lane–Emden–Matukuma equation, is to provide a natural interpolation of the Lane–Emden equation corresponding to the case \(\sigma =0\) and the Matukuma equation corresponding to the case \(\sigma =1\) . This is a continuation of our earlier work in which the sublinear case \(\alpha \le 1\) was studied. In the supercritical case, namely \(\alpha >1\) , we prove that the equation admits non-trivial, non-negative, \(C^2\) -solution if, and only if, \( \alpha > \frac{n+2-4\sigma }{n-2}. \) This provides a comprehensive overview of non-existence and existence results for the equation in the full generality of the parameters \(\sigma \in [0,1]\) and \(\alpha \in \textbf{R}\) .
