On the existence problem for a double mean field equation and its application to prescribed curvature problems

Abstract

Let $(M,g)$ be a compact Riemann surface with smooth boundary $\partial M$. Let $K(x)>0$, $h(x)>0$ or $h(x)\equiv 0$ be two smooth functions on $M$ and $\partial M$ respectively. In this paper, we consider the following double mean field type equation with a nonlinear Neumann boundary condition related to a prescribed curvature problem \begin{equation}\label{equation:01} \begin{cases} \displaystyle -\Delta_g u+\frac{4\pi}{|M|} =2\rho\frac{Ke^u}{\int_MKe^u\,dv_g},&\text{in }M,\\[1.2em] \displaystyle \partial_\nu u =2\rho'\frac{he^{u/2}}{\int_{\partial M}he^{u/2}\,ds_g},&\text{on }\partial M, \end{cases} \end{equation} where $\rho>0,\ \rho'\geq 0$ and $\rho+\rho'=2\pi$, $\nu$ is the outward unit normal vector to $\partial M$. When $h\equiv 0$, the second equation changes to $\partial_\nu u=0$ on $\partial M$. By introducing a boundary integration constant, we obtain a sufficient condition under which \eqref{equation:01} admits a solution. As a geometric application, we solve the prescribed curvature equation for both the Gauss curvature and the geodesic curvature in the case that $M$ is topologically a disk. When $(M,g)$ is the standard hemisphere $\mathbb S^2_+$, let $p\in \partial \mathbb S^2_+$ be a maximum point of $$ \Phi(q)=h(q)+\sqrt{h(q)^2+K(q)}, \qquad q\in\partial \mathbb S^2_+. $$ Suppose \begin{equation*} K_\nu(p)+2\Phi(p)(-\Delta_{\partial \mathbb S^2_+})^{1/2}h(p)<0. \end{equation*} Then the prescribed curvature problem \begin{align*} -\Delta_g u+\frac{4\pi}{|M|}=2Ke^u, \text{in }\mathbb S^2_+,\ \ \partial_\nu u=2he^{u/2}, \text{on }\partial \mathbb S^2_+. \end{align*} admits a smooth solution.