Non-existence and behaviour at infinity of solutions of some elliptic equations

When $\alpha\le 2\beta$, we will prove the non-existence of solutions of the equation $\Delta v+\alpha e^v+\beta (x\cdot\nabla v)e^v=0$ in $R^2$ which satisfy $\gamma =\int_{R^2}e^vdx/(2\pi) <\infty$ and $|x|^2e^{v(x)}\le C_1$ in $R^2$ for some constant $C_1>0$. When $\alpha>2\beta$, we will prove that if $v$ is a solution of the above equation, then there exist constants $0<\tau\le 1$ and $a_1$ such that $v(x)=-\gamma \log |x|+a_1+O(|x|^{-\tau})$ as $|x|\to\infty$ where $\gamma=(\alpha -2\beta)\gamma$. We will also show that $\gamma$ satisfies $\gamma>2$ and $\gamma<\alpha$.