On a doubly critical system involving fractional Laplacian with partial weight

In this paper, we establish a new improved Sobolev inequality based on a weighted Morrey space. To be precise, there exists $C=C(n,m,s,\alpha)>0$ such that for any $u,v \in {\dot{H}}^s(\mathbb{R}^{n})$ and for any $\theta \in (\bar{\theta},1)$, it holds that \begin{equation} \label{eq0.3} \Big( \int_{ \mathbb{R}^{n} } \frac{ |(uv)(y)|^{\frac{2^*_{s}(\alpha)}{2} } } { |y'|^{\alpha} } dy \Big)^{ \frac{1}{ 2^*_{s} (\alpha) }} \leq C ||u||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}} ||v||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}} ||(uv)||^{\frac{1-\theta}{2}}_{ L^{1,n-2s+r}(\mathbb{R}^{n},|y'|^{-r}) }, \end{equation} where $s \!\in\! (0,1)$, $0\!<\!\alpha\!<\!2s\!<\!n$, $2s\!<\!m\!<\!n$, $\bar{\theta}=\max \{ \frac{2}{2^*_{s}(\alpha)}, 1-\frac{\alpha}{s}\cdot\frac{1}{2^*_{s}(\alpha)}, \frac{2^*_{s}(\alpha)-\frac{\alpha}{s}}{2^*_{s}(\alpha)-\frac{2\alpha}{m}} \}$, $r=\frac{2\alpha}{ 2^*_{s}(\alpha) }$ 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.