Liouville-Type Theorems for Fractional and Higher-Order Hénon–Hardy Type Equations via the Method of Scaling Spheres

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*} $n>\alpha$, $0<\alpha<2$ or $\alpha=2m$ $1\leq m<\frac{n}{2}$. We first consider typical case $f(x,u)=|x|^{a}u^{p}$ $a\in(-\alpha,\infty)$ $00$, covered gap $p\in\big[\frac{n+\alpha+a}{n-\alpha},p_{c}(a)\big)$. particular, when $\alpha=2$, our give an affirmative answer conjecture posed by Phan Souplet \cite{PS}. As a consequence, derive priori estimates existence positive solutions Lane-Emden bounded domains all $1<p<\frac{n+2m}{n-2m}$. \cite{CFL,DPQ} remarkably maximal $p$. For $\Omega$, also apply spheres super-critical problems. Extensions PDEs IEs general nonlinearities $f(x,u)$ included. believe developed here can be applied conveniently various problems singularities without translation invariance cases moving planes conjunction Kelvin transforms do not work.