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

Tiêu đềOn the sub poly-harmonic property for solutions of (-Δ)^p u <0 in R^n
Loại công bốJournal Article
Năm xuất bản2017
Tác giảNgô, QAnh
Tạp chíComptes Rendus Mathématique
Tập355
Issue5
Trang526–532
Tóm tắt

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 $\mathbb R^n$.

URLhttp://dx.doi.org/10.1016/j.crma.2017.04.003
DOI10.1016/j.crma.2017.04.003