论文标题
在涉及部分重量的分数laplacian的双重关键系统上
On a doubly critical system involving fractional Laplacian with partial weight
论文作者
论文摘要
在本文中，我们基于加权莫雷空间建立了一种新的改进的Sobolev不平等。 To be precise, there exists $C=C(n,m,s,α)>0$ such that for any $u,v \in {\dot{H}}^s(\mathbb{R}^{n})$ and for any $θ\in (\barθ,1)$, it holds that \begin{equation} \label{eq0.3} \ big（\ int_ {\ mathbb {r}^{n}}} \ frac {|（uv）（y）|^{\ frac {2^*_ {s}（s}（α）}（α）} {2} {2}}}}}}}}}} {| y'|^α}} （α）}}} \ leq c || u || _ {\ dot {h}^s（\ mathbb {r}^{n}）}}^{\fracθ{2}}} || v || _ {\ dot {h}^s（\ mathbb {r}^{n}）}}^{\fracθ{2}}}}}} ||（uv）||^{\ frac {1- frac {1-θ}}} {2}} {2}} _ { l^{1，n-2s+r}（\ mathbb {r}^{n}，| y'|^{ - r}）}，\ end {equation {equation}其中$ s \！\ in \！ （0,1）$，$，$ 0 \！<\！α\！<\！<\！2S \！<\！n $，$ 2S \！<\！<\！<\！<\！ 1- \fracα{s} \ cdot \ frac {1} {2^*_ {s}（α）}，\ frac {2^*_ {s}（α） - \fracα{s}}}}} $ r = \ frac {2α} {2^*_ {s}（α）} $和$ y \！= \！（y'，y'，y'，y''）\ in \ mathbb {r}^{m} \ times \ times \ times \ times \ mathbb {r}^r}^n-m-m} $。 通过使用Mountain Pass引理和（\ ref {eq0.3}），我们为涉及$ \ Mathbb {r}^{n} $的分数laplacian的双重关键系统获得了一个非平凡的弱解决方案，并以直接的方式具有部分重量。此外，我们将不等式（\ ref {eq0.3}）扩展到更通用的形式，目的是研究某些具有部分重量的通用系统，尤其涉及p-laplacian。
In this paper, we establish a new improved Sobolev inequality based on a weighted Morrey space. To be precise, there exists $C=C(n,m,s,α)>0$ such that for any $u,v \in {\dot{H}}^s(\mathbb{R}^{n})$ and for any $θ\in (\barθ,1)$, it holds that \begin{equation} \label{eq0.3} \Big( \int_{ \mathbb{R}^{n} } \frac{ |(uv)(y)|^{\frac{2^*_{s}(α)}{2} } } { |y'|^α } dy \Big)^{ \frac{1}{ 2^*_{s} (α) }} \leq C ||u||_{\dot{H}^s(\mathbb{R}^{n})}^{\fracθ{2}} ||v||_{\dot{H}^s(\mathbb{R}^{n})}^{\fracθ{2}} ||(uv)||^{\frac{1-θ}{2}}_{ L^{1,n-2s+r}(\mathbb{R}^{n},|y'|^{-r}) }, \end{equation} where $s \!\in\! (0,1)$, $0\!<\!α\!<\!2s\!<\!n$, $2s\!<\!m\!<\!n$, $\barθ=\max \{ \frac{2}{2^*_{s}(α)}, 1-\fracα{s}\cdot\frac{1}{2^*_{s}(α)}, \frac{2^*_{s}(α)-\fracα{s}}{2^*_{s}(α)-\frac{2α}{m}} \}$, $r=\frac{2α}{ 2^*_{s}(α) }$ and $y\!=\!(y',y'') \in \mathbb{R}^{m} \times \mathbb{R}^{n-m}$. By using mountain pass lemma and (\ref{eq0.3}), we obtain a nontrivial weak solution to a doubly critical system involving fractional Laplacian in $\mathbb{R}^{n}$ with partial weight in a direct way. Furthermore, we extend inequality (\ref{eq0.3}) to more general forms on purpose of studying some general systems with partial weight, involving p-Laplacian especially.
