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

Abstract

AbstractIn this paper, we aim to develop the (direct) method of scaling spheres, its integral forms, and the method of scaling spheres in a local way. As applications, we investigate Liouville properties of nonnegative solutions to fractional and higher-order Hénon–Hardy type equations $$ \begin{align*}& (-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,u(x)) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n}, \,\,\, \mathbb{R}^{n}_{+} \,\,\, \text{or} \,\,\, B_{R}(0) \end{align*}$$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 Hénon–Hardy equations and equivalent integral equations (IEs). In $\mathbb {R}^{n}$, our results improve the known Liouville theorems for some especially admissible subranges of $a$ and $1<p<\min \big \{\frac {n+\alpha +a}{n-\alpha },p_{c}(a)\big \}$ to the full range $a\in (-\alpha ,\infty )$ and $p\in (0,p_{c}(a))$. In particular, when $a>0$, we covered the gap $p\in \big [\frac {n+\alpha +a}{n-\alpha },p_{c}(a)\big )$. For bounded domains (i.e., balls), 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 (Theorem 1.31). In addition to improving most of known Liouville type results to the sharp exponents in a unified way, 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.