Classification of solutions to the Liouville equation with a nonlinear Neumann boundary condition and applications to sharp inequalities

In this paper, we study the Liouville equation with a nonlinear Neumann boundary condition \begin{equation*} \begin{cases} -Δu = Ke^{2u} & $\text{in }$ $\mathbb{B}$^{2},\[2mm] \dfrac{\partial u}{\partial ν}+λ= ke^{u} & $\text{on }$ \partial$\mathbb{B}$^{2}, \end{cases} \end{equation*} where $K,k\in \mathbb{R}$, $\lambda \in (0,1]$ are constants, and $\nu$ denotes the outward unit normal on $\mathrm{\partial }{\mathbb{B}}^2$. We establish a classification of all smooth solutions to the equation. Our approach is a boundary-adapted P-function method which treats all signs of $K$ and $k$ within a unified framework. As applications of the classification result, we establish a family of sharp Sobolev-trace-type inequalities encompassing the classical Lebedev–Milin inequality, together with a corresponding deficit estimate.