Liouville type theorem for nonlinear elliptic equation with general nonlinearity

Xiaohui Yu

doi:10.3934/dcds.2014.34.4947

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2014, Volume 34, Issue 11: 4947-4966. Doi: 10.3934/dcds.2014.34.4947

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Liouville type theorem for nonlinear elliptic equation with general nonlinearity

Xiaohui Yu^1,

1.
The Center for China's Overseas Interests, Shenzhen University, Shenzhen Guangdong, 518060

Received: August 31, 2013

Revised: November 30, 2013

Published: October 2014

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Abstract

In this paper, we study the nonexistence of positive solutions for the following elliptic equation $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) & in \quad \mathbb{R}_+^N, \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) & on \quad \partial \mathbb{R}_+^N \end{array} \right. $$ and elliptic system $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u_1=f_1(u_1,u_2) &in \quad \mathbb{R}_+^N, \\ \\-\Delta u_2=f_2(u_1,u_2) & in\quad \mathbb{R}_+^N, \\ \displaystyle \\ \frac{\partial u_1}{\partial \nu}=g_1(u_1,u_2),\quad \frac{\partial u_2}{\partial \nu}=g_2(u_1,u_2) & on \quad \partial \mathbb{R}_+^N. \end{array} \right. $$ We will prove that these problems possess no positive solutions under some assumptions on nonlinear terms. The main technique we use is the moving plane method in an integral form.

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Mathematics Subject Classification: Primary: 35J60, 35J57; Secondary: 35J15.

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