论文标题
尖锐的逆转Hardy-littlewood-Sobolev不平等与扩展内核
Sharp reversed Hardy-Littlewood-Sobolev inequality with extended kernel
论文作者
论文摘要
在本文中，我们证明了以下相反的hardy-littlewood-sobolev不等式，具有扩展的内核\ begin {equination*} \ int _ {\ mathbb {r} _+^n} \ frac {x_n^β} {| x-y |^{n-α}} f（y） \ | g \ | _ {l^{Q'}（\ Mathbb {r} _+^n）} \ end end {qore {qore {qore*}对于任何非中性函数$ f \ in l^{p}（\ partial \ partial \ mathbb {r} l^{q'}（\ MathBb {r} _+^n）$，其中$ n \ geq2 $，$ p，\ q'\ in（0,1）$，$α> n $，$ 0 \leqββ<\leqβ<\ frac <\ frac {α-n} {α-n}} $ \ frac {n-1} {n} \ frac {1} {p}+\ frac {1} {q'} - \ frac {α+β-1} {n} = 1 $。我们证明了上述不平等的极端功能的存在。此外，在整形不变的情况下，我们通过移动球的变体方法来对所有极端函数进行分类，从而得出最佳常数，可以进行\ emph {而无需提升Lebesgue可测量溶液的规律性}。最后，我们通过使用pohozaev身份获得了充分的和必要的条件，以实现欧拉 - 拉格朗日方程的积极解决方案。我们的结果灵感来自Hang，Wang和Yan \ Cite {Hwy}，Dou，Guo和Zhu \ cite {DGZ}，$α<n $和$β= 1 $，以及Gluck \ cite {gl}，$α<n $ and $α<n $和$β\ geq0 $。
In this paper, we prove the following reversed Hardy-Littlewood-Sobolev inequality with extended kernel \begin{equation*} \int_{\mathbb{R}_+^n}\int_{\partial\mathbb{R}^n_+} \frac{x_n^β}{|x-y|^{n-α}}f(y)g(x) dydx\geq C_{n,α,β,p}\|f\|_{L^{p}(\partial\mathbb{R}_+^n)} \|g\|_{L^{q'}(\mathbb{R}_+^n)} \end{equation*} for any nonnegative functions $f\in L^{p}(\partial\mathbb{R}_+^n)$ and $g\in L^{q'}(\mathbb{R}_+^n)$, where $n\geq2$, $p,\ q'\in (0,1)$, $α>n$, $0\leqβ<\frac{α-n}{n-1}$, $p>\frac{n-1}{α-1-(n-1)β}$ such that $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}-\frac{α+β-1}{n}=1$. We prove the existence of extremal functions for the above inequality. Moreover, in the conformal invariant case, we classify all the extremal functions and hence derive the best constant via a variant method of moving spheres, which can be carried out \emph{without lifting the regularity of Lebesgue measurable solutions}. Finally, we derive the sufficient and necessary conditions for existence of positive solutions to the Euler-Lagrange equations by using Pohozaev identities. Our results are inspired by Hang, Wang and Yan \cite{HWY}, Dou, Guo and Zhu \cite{DGZ} for $α<n$ and $β=1$, and Gluck \cite{Gl} for $α<n$ and $β\geq0$.
