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Liouville-type theorems, radial symmetry and integral representation of solutions to Hardy-H\'enon equations involving higher order fractional Laplacians
Analysis of PDEs · Submitted 20 Sep 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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We study nonnegative solutions to the following Hardy-H\'enon type equations involving higher order fractional Laplacians $$ (-\Delta)^\sigma u = |x|^{-\alpha}u^{p} ~~~~~~ \mbox{in} ~ \mathbb{R}^n \backslash \{0\} $$ with a possible singularity at the origin, where $\sigma$ is a real number satisfying $0 < \sigma < n/2$, $-\infty < \alpha < 2\sigma$ 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 $\sigma \in (0, n/2)$ when $$ 1 < p < p_{\sigma,\alpha}^*:=\frac{n+2\sigma -2\alpha}{n-2\sigma} ~~~ \mbox{and} ~~~ -\infty < \alpha < 2\sigma. $$ This, in particular, covers a gap occurring for non-integral $\sigma \in (1, n/2)$ and $\alpha \in (0, 2\sigma)$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study nonnegative solutions to the following Hardy-H\'enon type equations involving higher order fractional Laplacians $$ (-\Delta)^\sigma u = |x|^{-\alpha}u^{p} ~~~~~~ \mbox{in} ~ \mathbb{R}^n \backslash \{0\} $$ with a possible singularity at the origin, where $\sigma$ is a real number satisfying $0 < \sigma < n/2$, $-\infty < \alpha < 2\sigma$ and $p>1$. | background 0.64 | 0.63 | 0.04 | asserted 0.82 |
| s1 | 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. | contribution 0.51 | 0.97 | 0.28 | asserted 1.00 |
| s2 | As the first application, we prove an optimal Liouville-type theorem for the above equations with removable singularity for all $\sigma \in (0, n/2)$ when $$ 1 < p < p_{\sigma,\alpha}^*:=\frac{n+2\sigma -2\alpha}{n-2\sigma} ~~~ \mbox{and} ~~~ -\infty < \alpha < 2\sigma. | result 0.95 | 0.96 | 0.36 | asserted 1.00 |
| s3 | $$ This, in particular, covers a gap occurring for non-integral $\sigma \in (1, n/2)$ and $\alpha \in (0, 2\sigma)$ in the current literature. | gap 0.56 | 0.44 | 0.07 | asserted 0.99 |
| s4 | 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. | result 0.83 | 0.96 | 0.19 | asserted 1.00 |
| s5 | Such radial symmetry would be useful in studying the singular Yamabe-type problems. | interpretation 0.86 | 0.20 | 0.04 | hedged possibility 0.82 |
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Original abstract
We study nonnegative solutions to the following Hardy-H\'enon type equations involving higher order fractional Laplacians $$ (-\Delta)^\sigma u = |x|^{-\alpha}u^{p} ~~~~~~ \mbox{in} ~ \mathbb{R}^n \backslash \{0\} $$ with a possible singularity at the origin, where $\sigma$ is a real number satisfying $0 < \sigma < n/2$, $-\infty < \alpha < 2\sigma$ 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 $\sigma \in (0, n/2)$ when $$ 1 < p < p_{\sigma,\alpha}^*:=\frac{n+2\sigma -2\alpha}{n-2\sigma} ~~~ \mbox{and} ~~~ -\infty < \alpha < 2\sigma. $$ This, in particular, covers a gap occurring for non-integral $\sigma \in (1, n/2)$ and $\alpha \in (0, 2\sigma)$ 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.
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4 arXiv papers cite this in 5 sentences
Papers citing it · 4 sentences without a confident label
Uses its method or tool 1 paper
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Finally, applying the iteration method in [ref] , we obtain the desired estimates ([ref]) and ([ref]).
Liouville theorems for conformal $Q$-curvature equations · Equivalence of the PDE and its integral representation · 1.00
What it cites · 5 sentences without a confident label
Uses the method or tool of 1 paper
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Here we will prove Theorem [ref] by using the method of moving spheres in Li, Zhang and Zhu [ref] and a “Bootstrap" iteration argument in Dai-Qin [ref] .
Liouville type theorems for fractional and higher order H\'{e}non-Hardy type equations via the method of scaling spheres · Liouville-type theorems · 0.99
Cites as background 1 paper
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One may also see [ref] and the references therein for some earlier partial results.
Liouville type theorems, a priori estimates and existence of solutions for non-critical higher order Lane-Emden-Hardy equations · Introduction and main results · 0.93
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