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