Nonlinear Neumann boundary problems for $n$-Laplacian Liouville equation on a half space

In this paper, for general $n\geq2$, we classify solutions to $n$-Laplacian Liouville equation with positive nonlinear Neumann boundary condition on the half-space $\mathbb{R}^{n}_{+}$. Under the positive nonlinear Neumann boundary condition, our result extend the classification result for the second order Liouville equation in \cite{Li} from $n=2$ to general $n\geq2$, and also extend the classification result for critical $p$-Laplacian equation in \cite{Zhou} from $p<n$ to $p=n$.

💡 Research Summary

The paper addresses the classification of solutions to the n‑Laplacian Liouville equation posed on the half‑space (\mathbb{R}^{n}_{+}) with a positive nonlinear Neumann boundary condition: \