Classification of solutions to the Liouville equation with a nonlinear Robin boundary condition on the unit disk

In this paper, we study the nonlinear boundary value problem \begin{equation*} \begin{cases} -Δu=e^{2u},& \mbox{in } {$\mathbb{D}$},\ \frac{\partial u}{\partialν}+λ=e^u ,& \mbox{on } {$\mathbb{S}$^{1}}, \end{cases} \end{equation*} where $\mathbb{D}$ is the unit disk, $\lambda$ is a constant and $\nu$ denotes the outer unit normal on ${\mathbb{S}}^1$. For $0<\lambda \le 2$, we establish a complete classification of smooth solutions. For $2<\lambda \le 3$, we prove a dichotomy between radial solutions and nonradial solutions. We develop a Hardy-Wronskian boundary rigidity method that transforms the nonlinear boundary value problem into a spectral rigidity problem for normalized holomorphic frames. This method exploits the complex-analytic structure of the Liouville equation with a Robin boundary condition and provides a new rigidity framework for related two-dimensional elliptic boundary value problems.