Title | On the sub poly-harmonic property for solutions of (-Δ)^p u <0 in R^n |

Publication Type | Journal Article |

Year of Publication | 2017 |

Authors | Ngô, QAnh |

Journal | Comptes Rendus Mathématique |

Volume | 355 |

Issue | 5 |

Pagination | 526–532 |

Abstract | In this note, we mainly study the relation between the sign of $(-\Delta)^p u$ and $(-\Delta)^{p-i} u$ in $\mathbb R^n$ with $p \geqslant 2$ and $n \geqslant 2$ for $1 \leqslant i \leqslant p-1$. Given the differential inequality $(-\Delta)^p u < 0$, first we provide several sufficient conditions so that $(-\Delta)^{p-1} u < 0$ holds. Then we provide conditions such that $(-\Delta)^i u < 0$ for all $i=1,2,...,p-1$ which is known as the sub poly-harmonic property for $u$. In the last part of the note, we revisit the super poly-harmonic property for solutions of $(-\Delta)^p u = e^{2pu}$ and $(-\Delta)^p u = u^q$ with $q>0$ in $\mathbb R^n$. |

URL | http://dx.doi.org/10.1016/j.crma.2017.04.003 |

DOI | 10.1016/j.crma.2017.04.003 |