Author

Xiao-Han Cai

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Preprint Sep 2026

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} -\Delta u = Ke^{2u}&\text{in } \mathbb{B}^{2},\\[2mm] \dfrac{\partial u}{\partial \nu}+\lambda = ke^{u}&\text{on } \partial\mathbb{B}^{2}, \end{cases} \end{equation*} where $K,k\in\mathbb{R}$, $\la...

Xiao-Han Cai, Abdolhakim Shouman · 0 citations

Preprint Jul 2026

Sharp rigidity for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature

We study the quasilinear Liouville equation \[ -\Delta_n u=e^u \] on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, if a solution $u$ satisfies the optimal logarithmic lower bound \[ u(x)\ge -\frac{n^2}{n-1}\log r(x)+o(\log r(x)) \quad \text{as }r(x)\to+\infty, \...

Xiaohan Cai · 0 citations