Liouville-type theorems, radial symmetry and integral representation of solutions to Hardy-Hénon equations involving higher order fractional Laplacians

Abstract

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.

Theorems & Definitions (39)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Corollary 1.8
• Lemma 2.1
• Proposition 2.2