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