Liouville-type theorems, radial symmetry and integral representation of solutions to Hardy-Hénon equations involving higher order fractional Laplacians
Hui Yang
备注：
39 pages. Added an existence result in the supercritical case
We study nonnegative solutions to the following Hardy-Hénon type equations involving higher order fractional Laplacians $$ (-Δ)^σu = |x|^{-α}u^{p} ~~~~~~ \mbox{in} ~ \mathbb{R}^n \backslash \{0\} $$ with a possible singularity at the origin, where $σ$ is a real number satisfying $0 < σ< n/2$, $-\infty < α< 2σ$ and $p>1$. By a more direct approach without using the super poly-harmonic properties, we establish an integral representation for nonnegative solutions to the above higher order fractional equations whether the singularity $\{0\}$ is removable or not. As the first application, we prove an optimal Liouville-type theorem for the above equations with removable singularity for all $σ\in (0, n/2)$ when $$ 1 < p < p_{σ,α}^*:=\frac{n+2σ-2α}{n-2σ} ~~~ \mbox{and} ~~~ -\infty < α< 2σ. $$ This, in particular, covers a gap occurring for non-integral $σ\in (1, n/2)$ and $α\in (0, 2σ)$ in the current literature. As the second application, we show the radial symmetry of solutions in the critical case or in the case when the origin is a non-removable singularity. Such radial symmetry would be useful in studying the singular Yamabe-type problems.
In this paper, we are concerned with the critical order Lane-Emden-Hardy equations \begin{equation*} (-Δ)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq4$ is even, $0\leq a<n$ and $1<p<+\infty$. We prove Liouville theorem for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the unique nonnegative solution is $u\equiv0$. Our result seems to be the first Liouville theorem on the critical order equations in higher dimensions ($n\geq3$).
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In this paper, we consider the critical order Hardy-Hénon equations \begin{equation*} (-Δ)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}}, \,\,\,\,\,\,\,\,\,\,\, x \, \in \,\, \mathbb{R}^{n}, \end{equation*} where $n\geq4$ is even, $-\infty<a<n$, and $1<p<+\infty$. We first prove a Liouville theorem (Theorem \ref{Thm0}), that is, the unique nonnegative solution to this equation is $u\equiv0$. Then as an immediate application, we derive a priori estimates and hence existence of positive solutions to critical order Lane-Emden equations in bounded domains (Theorem \ref{Thm1} and \ref{Thm2}). Our resu…
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