Work

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

Year: 2024

Type: article

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 x n . Based on this, we prove the non-existence of non-negative solutions.

Source: Proceedings of the Royal Society of Edinburgh Section A Mathematics

DOI: 10.1017/prm.2024.98

Author: Huxiao Luo

Institutions: Zhejiang Normal University, Shanghai Jiao Tong University

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: National Natural Science Foundation of China, Shanghai Jiao Tong University

Award: G8548478608