Liouville type theorem for nonlinear elliptic equation with general nonlinearity
Liouville type theorem for nonlinear elliptic equation with general nonlinearity
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DOI：
10.3934/dcds.2014.34.4947
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发表时间：
2014-05
影响因子：
1.1
通讯作者：
Xiaohui Yu
中科院分区：
文献类型：
--
作者：
Xiaohui Yu
本文研究了如下椭圆型方程$$Left\Begin{ARRAY}{ll}\DisplayStyle-\Delta u=f(U)&in\quad\mathbb{R}_+^N，\displaystyle\\frac{\Partial u}{\Partial\nu}=g(U)&on\quad\Partial\Mathbb{R}+^N\end{ARRAY}。$$和椭圆系统$$\Left\\Begin{ARRAY}{ll}\DisplayStyle-\Delta u_1=f_1(u_1，u_2)\quad\mathbb{R}_+^N，\-\Delta u_2=f_2(u_1，u_2)&in\quad\mathbb{R}_+^N，\displaystyle\\Frac{\Partial u_1}{\Partial\nu}=g_1(u_1，u_2)，\quad\frc{\Partial u_2}{\Partial\nu}=g_2(u_1，u_2)&on\quad\Partial\mathbb{R}_+^N。在非线性项的某些假设下，我们将证明这些问题不存在正解。我们使用的主要技术是积分形式的移动平面法。
In this paper, we study the nonexistence of positive solutions for the following elliptic equation $$ \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) \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.