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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 $\gamma<\alpha$.