Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians

Cao, Daomin; Dai, Wei; Qin, Guolin

doi:10.1090/tran/8389

Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians
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by Daomin Cao, Wei Dai and Guolin Qin PDF

Trans. Amer. Math. Soc. 374 (2021), 4781-4813 Request permission

Abstract:

In this paper, we are concerned with the following equations \begin{equation*} \\\begin {cases} (-\Delta )^{m+\frac {\alpha }{2}}u(x)=f(x,u,Du,\cdots ), x\in \mathbb {R}^{n}, \\ u\in C^{2m+[\alpha ],\{\alpha \}+\epsilon }_{loc}\cap \mathcal {L}_{\alpha }(\mathbb {R}^{n}), u(x)\geq 0, x\in \mathbb {R}^{n} \end{cases}\end{equation*} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to the above equations. Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to the above fractional higher-order equations with general nonlinearities $f(x,u,Du,\cdots )$ including conformally invariant and odd order cases. In particular, we classify nonnegative classical solutions to all odd order conformally invariant equations. Our results completely improve the classification results for third order conformally invariant equations in Dai and Qin (Adv. Math., 328 (2018), 822-857) by removing the assumptions on integrability. We also give a crucial characterization for $\alpha$-harmonic functions via outer-spherical averages in the appendix.

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