Liouville type theorems, a priori estimates and existence of solutions for non-critical higher order Lane-Emden-Hardy equations

Abstract

In this paper, we are concerned with the non-critical higher order Lane-Emden-Hardy equations \begin{equation*} (-Δ)^{m}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\ge 3$, $1\le m<\frac{n}{2}$, $0\le a<2m$, $1<p<\frac{n+2m-2a}{n-2m}$ if $0\le a<2$, and $1<p<\infty$ if $2\le a<2m$. We prove Liouville theorems for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the unique nonnegative solution is $u\equiv 0$. As an application, we derive a priori estimates and existence of positive solutions to non-critical higher order Lane-Emden equations in bounded domains (Theorem \ref{Thm1} and \ref{Thm2}). The results for critical order Hardy-Hénon equations have been established by Chen, Dai and Qin \cite{CDQ} recently.

arXiv admin note: substantial text overlap with arXiv:1808.06609; text overlap with arXiv:1808.01581