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 $\partial M$. 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 rel