In this paper, we classify the solutions to the Liouville equation with Neumann boundary on the unit disc \(\overline{\mathbb {B}^2}\) as follows \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=e^{2u},& \text{ in } {\mathbb {B}^2},\\ \frac{\partial u}{\partial \nu }+\lambda =0 ,& \text{ on } {\mathbb {S}^{1}}, \end{array}\right. } \end{aligned}\) where \(\lambda \in \mathbb {R}\) , \(\nu \) is the outer unit normal on \(\mathbb {S}^{1}\) . The equation is related to an Onofri-type inequality with Neumann boundary on \(\mathbb {B}^2\) . Employing some important integral identities with the Obata identity, we prove the classification results to the above equation for \(0<\lambda \le 1\) .
