半空间上Choquard方程的Liouville型定理

Liouville-Type Theorem for Choquard Equation in Half-Space

摘要: 本文研究半空间上Choquard方程$\{\begin{array}{l}-\Delta u\left(y\right)={\displaystyle {\int }_{\partial {ℝ}_{+}^N}\frac{{\left|u\left(\overline{x},0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}\overline{x}}{\left|u\left(y\right)\right|}^{p-2}u\left(y\right),y\in {ℝ}_{+}^N\\ \frac{\partial u}{\partial \nu }\left(\overline{x},0\right)={\displaystyle {\int }_{{ℝ}_{+}^N}\frac{{\left|u\left(y\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}y}{\left|u\left(\overline{x},0\right)\right|}^{p-2}u\left(\overline{x},0\right),\left(\overline{x},0\right)\in \partial {ℝ}_{+}^N\end{array}$应用积分形式的移动平面法，证明了在参数p的一定取值条件下，该方程不存在非平凡正解。

Abstract: This paper investigates the Choquard equation in a half-space setting $\{\begin{array}{l}-\Delta u\left(y\right)={\displaystyle {\int }_{\partial {ℝ}_{+}^N}\frac{{\left|u\left(\overline{x},0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}\overline{x}}{\left|u\left(y\right)\right|}^{p-2}u\left(y\right),y\in {ℝ}_{+}^N\\ \frac{\partial u}{\partial \nu }\left(\overline{x},0\right)={\displaystyle {\int }_{{ℝ}_{+}^N}\frac{{\left|u\left(y\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}y}{\left|u\left(\overline{x},0\right)\right|}^{p-2}u\left(\overline{x},0\right),\left(\overline{x},0\right)\in \partial {ℝ}_{+}^N\end{array}$. By applying the method of moving planes in integral form, it is demonstrated that under certain conditions on the parameter p, the equation has no nontrivial positive solutions.