Liouville type theorem for critical order Lane-Emden-Hardy equations in $\mathbb{R}^n$

TL;DR

This paper establishes a Liouville theorem for nonnegative solutions of critical order Lane-Emden-Hardy equations in higher dimensions, proving that the only solution is the trivial zero function.

Contribution

It provides the first Liouville theorem for critical order equations in dimensions $n geq 3$, extending the understanding of such equations.

Findings

Nonnegative solutions are trivial (zero) in the considered equations.

The theorem applies to even dimensions $n geq 4$ with specific parameters.

First such Liouville theorem for critical order equations in higher dimensions.

Abstract

In this paper, we are concerned with the critical order Lane-Emden-Hardy equations \begin{equation*} (-\Delta)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq4$ is even, $0\leq a<n$ and $1<p<+\infty$. We prove Liouville theorem for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the unique nonnegative solution is $u\equiv0$. Our result seems to be the first Liouville theorem on the critical order equations in higher dimensions ($n\geq3$).