  In this note, we study Liouville type theorem for conformal Gaussian
curvature equation (also called the mean field equation) $$ -\Delta u=K(x)e^u,
in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a
sign-changing smooth function in the real line $R$, we have a non-existence
result for the finite total curvature solutions. When $K$ is monotone
non-decreasing along every ray starting at origin, we can prove a non-existence
result too. We use moving plane method and moving sphere method.
