On the classification of solutions to a class of $N$-Liouville equations in $\mathbb{R}^N$
Given $N\geq 2$ and $\alpha>-1$, we consider the following weighted Liouville-type equation involving the $N$-Laplacian: \begin{equation*} \left\{ \begin{aligned} -& \Delta_N u = |x|^{N\alpha} e^u \quad \text{ in } \mathbb{R}^N && , \\ & \int_{\mathbb{R}^N} |x|^{N\alpha} e^u \, dx < + \infty\,. &&\end{aligned} \right. \end{equation*} Solutions have been completely classified when $N=2$ via complex analysis, and when $\alpha=0$ using Pohozaev identities and an isoperimetric argument. In this paper, we first devise a $P$-function approach to the classification result for all $\alpha>-1$ when $N=2$. Since it is not based on complex analysis, this alternative and more PDE-oriented approach naturally extends to $N\geq 3$ by providing the classification for any $-1<\alpha\leq 0$. In particular, the explicit radial solutions are the unique ones for $-1<\alpha\leq0$ but become degenerate for special values $\alpha_k>0$, a hint that non-radial solutions might arise for $\alpha>0$ as it happens when $N=2$.