  The problem of prescribing Gaussian curvature on Riemann surface with conical
singularity is considered. Let $(\Sigma,\beta)$ be a closed Riemann surface
with a divisor $\beta$, and $K_\lambda=K+\lambda$, where
$K:\Sigma\rightarrow\mathbb{R}$ is a H\"older continuous function satisfying
$\max_\Sigma K= 0$, $K\not\equiv 0$, and $\lambda\in\mathbb{R}$. If the Euler
characteristic $\chi(\Sigma,\beta)$ is negative, then by a variational method,
it is proved that there exists a constant $\lambda^\ast>0$ such that for any
$\lambda\leq 0$, there is a unique conformal metric with the Gaussian curvature
$K_\lambda$; for any $\lambda$, $0<\lambda<\lambda^\ast$, there are at least
two conformal metrics having $K_\lambda$ its Gaussian curvature; for
$\lambda=\lambda^\ast$, there is at least one conformal metric with the
Gaussian curvature $K_{\lambda^\ast}$; for any $\lambda>\lambda^\ast$, there is
no certain conformal metric having $K_{\lambda}$ its Gaussian curvature. This
result is an analog of that of Ding and Liu \cite{Ding-Liu}, partly resembles
that of Borer, Galimberti and Struwe \cite{B-G-Stru}, and generalizes that of
Troyanov \cite{Troyanov} in the negative case.
