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 $K,k\in\mathbb{R}$, $λ\in(0,1]$ are constants, and $ν$ denotes the outward unit normal on $ \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.