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 $s\text{\hspace{0.17em}⁣}\in \text{\hspace{0.17em}⁣}(0,1)$, $0\text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}\alpha \text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}2s\text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}n$, $2s\text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}m\text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}n$, $\bar{\theta}=\max { \frac{2}{2^_{s}(\alpha)}, 1-\frac{\alpha}{s}\cdot\frac{1}{2^{s}(\alpha)}, \frac{2^*{s}(\alpha)-\frac{\alpha}{s}}{2^_{s}(\alpha)-\frac{2\alpha}{m}} }$,$r=\frac{2\alpha}{ 2^_{s}(\alpha) }$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.