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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

Shu-Yu Hsu 1,

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 $\gamma<\alpha$.
Keywords: singular diffusion equation., asymptotic expansion, behaviour at infinity, Non-existence of solution.
Mathematics Subject Classification: Primary: 35B05; Secondary: 35J6.
Citation: Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769
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