Non-existence and behaviour at infinity of solutions of some elliptic equations (Q1433453)

The author studies the solutions of the equation \[ \Delta v+\alpha e^v+ \beta(x\cdot\nabla v)e^v= 0\quad\text{in }\mathbb{R}^2,\tag{1} \] where \(\alpha\), \(\beta\) are constants. He shows that when \(\alpha\leq2\beta\) there does not exist any solution of (1) satisfying \(\int_{\mathbb{R}^2} e^v \,dx< \infty\) and \(| x|^2 e^{v(x)}\leq C_1\) \(\forall x\in\mathbb{R}^2\) for some constant \(C_1> 0\). When \(\alpha> 2\beta\) the author obtains an asymptotic expansion for the solution of (1) satisfying the above conditions.