Large conformal metrics with prescribed Gaussian and geodesic curvatures

We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on unit disk. This is equivalent to solving following P.D.E. $$\begin{aligned} {\left\{ \begin{array}{ll}-\Delta u=2K(z)e^u&{}\hbox {in}\;\mathbb {D}^2,\\ \partial _\nu u+2=2h(z)e^\frac{u}{2}&{}\hbox {on}\;\partial \mathbb {D}^2,\end{array}\right. } \end{aligned}$$ where K, h are prescribed curvatures. construct family metrics with $$K_\varepsilon ,h_\varepsilon $$ converging respectively as $$\varepsilon goes 0, which blows up at one boundary point under some generic assumptions.