半空间上Hartree方程的Liouville型定理

摘要/Abstract

摘要：

该文研究半空间上的Hartree方程 $ \left\{ \begin{array}{l} \displaystyle -\Delta u_i(y)=\sum\limits_{j=1}^n\int_{\partial\mathbb{R} _+^N} \frac{ F(u_j(\bar x, 0))}{|(\bar x, 0)-y|^{N-\alpha}}{\rm d}\bar xg(u_i(y)), &y\in\mathbb{R} _+^N, \\\displaystyle \frac{\partial u_i}{\partial \nu}(\bar x, 0)=\sum\limits_{j=1}^N\int_{\mathbb{R} _+^N} \frac{G(u_j(y))}{|(\bar x, 0)-y|^{N-\alpha}}\, {\rm d}y f(u_i(\bar x, 0)), &(\bar x, 0)\in\partial \mathbb{R} _+^N. \end{array} \right. $ 非平凡正解的非存在性结果. 在对非线性项F, G, f, g作适当的假设下, 证明上述方程只有常数的正解. 该文用积分形式的移动平面法来证明.

关键词: Hartree方程, 半空间, 移动平面法, Liouville型定理

Abstract:

In this paper, we study the nonexistence of nontrivial positive solution for the following Hartree equation in half space $ \left\{ \begin{array}{l} \displaystyle -\Delta u_i(y)=\sum\limits_{j=1}^n\int_{\partial\mathbb{R} _+^N} \frac{ F(u_j(\bar x, 0))}{|(\bar x, 0)-y|^{N-\alpha}}{\rm d}\bar xg(u_i(y)), &y\in\mathbb{R} _+^N, \\\displaystyle \frac{\partial u_i}{\partial \nu}(\bar x, 0)=\sum\limits_{j=1}^N\int_{\mathbb{R} _+^N} \frac{G(u_j(y))}{|(\bar x, 0)-y|^{N-\alpha}}\, {\rm d}y f(u_i(\bar x, 0)), &(\bar x, 0)\in\partial \mathbb{R} _+^N. \end{array} \right. $ Under some assumptions on the nonlinear functions F, G, f, g, we will show that the positive solutions of the above equation must be constants. We use moving plane method in an integral form to prove our result.

Key words: Hartree equation, Half space, Moving plane method, Liouville type theorem

中图分类号:

O175.25

李泓桥. 半空间上Hartree方程的Liouville型定理[J]. 数学物理学报, 2021, 41(2): 388-401.

Hongqiao Li. Liouville Type Theorem for Hartree Equations in Half Spaces[J]. Acta mathematica scientia,Series A, 2021, 41(2): 388-401.

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