July  2004, 10(3): 769-786. doi: 10.3934/dcds.2004.10.769

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

1. 

School of Mathematics, University of Minnesota–Twin Cities, Minneapolis, MN 55455, United States

Received  August 2002 Revised  July 2003 Published  January 2004

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 $\gamma<\alpha$.
Citation: Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769
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