A Liouville type theorem for higher order Hardy-Hénon equation in Rⁿ

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• 作者：Tingzhi Cheng^a ; ^{nowitzki1989@126.com" class="auth_mail" title="E-mail the corresponding author} ; Shuang Liu^b ; ^{shuang@math.sc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Liouville theorems ; Higher order Hardy&ndash ; Hé ; non equations ; Integral equations ; Method of moving planes in integral forms
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：444
• 期：1
• 页码：370-389
• 全文大小：398 K

文摘

In this paper, we consider the following higher order Hardy–Hénon type equations in Rⁿden">de">$R^n$: in subcritical cases with a>0den">de">$a>0$, and in particular, we focus on the non-existence of positive solutions.

First, under some very mild growth conditions, we show that problem (1) is equivalent to the integral equation

where G(x,y)den">de">$G(x,y)$ is the Green's function associated with de288007b0" title="Click to view the MathML source">(−Δ)^mden">de">${(-\mathrm{\Delta })}^m$ in Rⁿden">de">$R^n$.

Then by using the method of moving planes in integral forms, we prove that there is no positive solution for integral equation (2) in subcritical cases 160" class="mathmlsrc">160.gif&_user=111111111&_pii=S0022247X16301883&_rdoc=1&_issn=0022247X&md5=9848f8803f82be1507a541a986b98ad1">160.gif">den">de">. For the non-existence of positive radially symmetric solutions, we can extend the range to subcritical cases den">de">$1<p<\frac{n+2m+2a}{n-2m}$. This partially solves an open conjecture posed by Quoc Hung Phan and Philippe Souplet [21].