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

Let $(M,g)$ be a compact Riemann surface with smooth boundary $\mathrm{\partial }M$. Let $K(x)>0$, $h(x)>0$ or $h(x)\equiv 0$ be two smooth functions on $M$ and $\mathrm{\partial }M$ 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 -Δ_g u+\frac{4π}{|M|} =2ρ\frac{Ke^u}{\int_MKe^u,dv_g}, & $\text{in }$M,\[1.2em] \displaystyle \partial_νu =2ρ’\frac{he^{u/2}}{\int_{\partial M}he^{u/2},ds_g}, & $\text{on }$\partial M, \end{cases} \end{equation} where $\rho >0,{\rho }^{\mathrm{\prime }}\ge 0$ and $\rho +{\rho }^{\mathrm{\prime }}=2\pi$, $\nu$ is the outward unit normal vector to $\mathrm{\partial }M$. When $h\equiv 0$, the second equation changes to ${\mathrm{\partial }}_{\nu }u=0$ on $\mathrm{\partial }M$. 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 $p\in \mathrm{\partial }{\mathbb{S}}_{+}^2$ be a maximum point of $$\mathrm{\Phi }(q)=h(q)+\sqrt{h(q{)}^2+K(q)},q\in \mathrm{\partial }{\mathbb{S}}_{+}^2\mathrm{.}$$ Suppose \begin{equation*} K_ν(p)+2Φ(p)(-Δ_{\partial \mathbb S^2_+})^{1/2}h(p)<0. \end{equation*} Then the prescribed curvature problem \begin{align*} -Δ_g u+\frac{4π}{|M|}=2Ke^u, $\text{in }$\mathbb S^2_+,\ \ \partial_νu=2he^{u/2}, $\text{on }$\partial \mathbb S^2_+. \end{align*} admits a smooth solution.