Large Conformal metrics with prescribed sign-changing Gauss curvature

Let $(M,g)$ be a two dimensional compact Riemannian manifold of genus $g(M)>1$. Let $f$ be a smooth function on $M$ such that $$f\ge 0,f\equiv ̸0,\underset{M}{\min }f=0.$$ Let $p_1,\dots ,p_n$ be any set of points at which $f(p_i)=0$ and $D^{2}f(p_i)$ is non-singular. We prove that for all sufficiently small $\lambda >0$ there exists a family of “bubbling” conformal metrics $g_{\lambda }=e^{u_{\lambda }}g$ such that their Gauss curvature is given by the sign-changing function $K_{g_{\lambda }}=-f+{\lambda }^2$. Moreover, the family $u_{\lambda }$ satisfies $$u_{\lambda }(p_j)=-4\log \lambda -2\log (\frac{1}{\sqrt{2}}\log \frac{1}{\lambda })+O(1)$$ and $${\lambda }^2{e}^{u_{\lambda }}\rightharpoonup 8\pi \sum _{i=1}^n{\delta }_{p_i},as\lambda \to 0,$$ where ${\delta }_p$ designates Dirac mass at the point $p$.