Liouville type theorems for fractional and higher order H\'{e}non-Hardy type equations via the method of scaling spheres

Abstract

In this paper, we are concerned with the fractional and higher order H\'{e}non-Hardy type equations \begin{equation*} (-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,u(x)) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n}, \,\,\, \mathbb{R}^{n}_{+} \,\,\, \text{or} \,\,\, \Omega \end{equation*} with $n>\alpha$, $0<\alpha<2$ or $\alpha=2m$ with $1\leq m<\frac{n}{2}$. We first consider the typical case $f(x,u)=|x|^{a}u^{p}$ with $a\in(-\alpha,\infty)$ and $0<p<p_{c}(a):=\frac{n+\alpha+2a}{n-\alpha}$. By using the method of scaling spheres, we prove Liouville theorems for the above H\'{e}non-Hardy equations and equivalent integral equations in $\mathbb{R}^{n}$ and $\mathbb{R}^{n}_{+}$. Our results improve the known Liouville theorems for some especially admissible subranges of $a$ and $1<p<\min\left\{\frac{n+\alpha+a}{n-\alpha},p_{c}(a)\right\}$ to the full range $a\in(-\alpha,\infty)$ and $p\in(0,p_{c}(a))$. When $a>0$, we covered the gap $p\in\big[\frac{n+\alpha+a}{n-\alpha},p_{c}(a)\big)$. In particular, when $\alpha=2$, our results give an affirmative answer to the conjecture posed by Phan and Souplet \cite{PS}. As a consequence, we derive a priori estimates and existence of positive solutions to higher order Lane-Emden equations in bounded domains for all $1<p<\frac{n+2m}{n-2m}$. Our theorems improve the results in \cite{CFL,DPQ} remarkably to the maximal range of $p$. For bounded domains $\Omega$, we also apply the method of scaling spheres to derive Liouville theorems for super-critical problems. Extensions to PDEs and IEs with general nonlinearities $f(x,u)$ are also included. We believe the method of scaling spheres developed here can be applied conveniently to various fractional or higher order problems with singularities or without translation invariance or in the cases the method of moving planes in conjunction with Kelvin transforms do not work.