Liouville theorems for the sub-linear Lane–Emden equation on the half space

Abstract

In this article, we study the following Dirichlet problem to the sub-linear Lane–Emden equation \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=u^{p},\quad u(x)\geq0,\quad x\in\mathbb{R}^n_+, \\ u(x)\equiv0,\quad x\in\partial\mathbb{R}^n_+, \end{array} \right. \end{equation*} where $n\geq3$ , $0 \lt p\leq1$ . By establishing an equivalent integral equation, we give a lower bound of the Kelvin transformation $\bar{u}$ . Then, by constructing a new comparison function, we apply the maximum principle based on comparisons and the method of moving planes to obtain that u only depends on xn. Based on this, we prove the non-existence of non-negative solutions.