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Liouville theorems for the sub-linear Lane–Emden equation on the half space
Part of:
Partial differential equations
Published online by Cambridge University Press: 18 November 2024
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.
MSC classification
Primary:
35A01: Existence problems
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- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
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