Liouville type theorem for Hartree-Fock Equation on half space

Abstract

In this paper, we study the Liouville type theorem for the following Hartree-Fock equation in half space

\begin{document}$ \begin{align*} \begin{cases} - \Delta {u_i}(y) = \sum\limits_{j = 1}^n {{\int _{\partial \mathbb{R}_ + ^N}}} \frac{{{u_j}(\bar x, 0){F_1}({u_j}(\bar x, 0))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}d\bar x{f_2}({u_i}(y)) \\ \qquad \qquad \qquad + \sum\limits_{j = 1}^n {{\int _{\partial \mathbb{R}_ + ^N}}} \frac{{{u_j}(\bar x, 0){F_2}({u_i}(\bar x, 0))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}d\bar x{f_1}({u_j}(y)), \ y \in \mathbb{R}_ + ^N, \hfill \\ \frac{{\partial {u_i}}} {{\partial \nu }}(\bar x, 0) = \sum\limits_{j = 1}^n {{\int _{ \mathbb{R}_ + ^N}}} \frac{{{u_j}(y){G_1}({u_j}(y))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}dy{g_2}({u_i}(\bar x, 0)) \\ \qquad \qquad \qquad + \sum\limits_{j = 1}^n {{\int _{ \mathbb{R}_ + ^N}}} \frac{{{u_j}(y){G_2}({u_i}(y))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}dy{g_1}({u_j}(\bar x, 0)), \quad \quad(\bar x, 0) \in \partial \mathbb{R}_ + ^N, \end{cases} \end{align*} $\end{document}

where \begin{document}$ \mathbb{R}_+^N = \{x\in{\mathbb{R}^N}: x_N > 0\}, f_1, f_2, g_1, g_2, F_1, F_2, G_1, G_2 $\end{document} are some nonlinear functions. Under some assumptions on the nonlinear functions \begin{document}$ F, G, f, g $\end{document}, we will prove the above equation only possesses trivial positive solution. We use the moving plane method in an integral form to prove our result.