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

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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

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

Received August 2002 Revised July 2003 Published January 2004

Abstract
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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 & Continuous Dynamical Systems - A, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769

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