Every nonnegative solution of the static Schrödinger-Hartree-Maxwell equations with higher-order or fractional Laplacians is either zero, or in the critical case an explicit rescaled bubble.
Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
In this paper, we are concerned with equations \eqref{PDE} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to \eqref{PDE} (Theorem \ref{Thm0}). 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 fractional higher-order equations \eqref{PDE} with general nonlinearities $f(x,u,Du,\cdots)$ including conformally invariant and odd order cases. In particular, our results completely improve the classification results for third order equations in Dai and Qin \cite{DQ1} by removing the assumptions on integrability. We also derive a characterization for $\alpha$-harmonic functions via averages in the appendix.
citation-role summary
background 1
citation-polarity summary
fields
math.AP 1years
2019 1verdicts
CONDITIONAL 1roles
background 1polarities
background 1representative citing papers
citing papers explorer
-
Classification of nonnegative solutions to static Schr\"{o}dinger-Hartree-Maxwell type equations
Every nonnegative solution of the static Schrödinger-Hartree-Maxwell equations with higher-order or fractional Laplacians is either zero, or in the critical case an explicit rescaled bubble.