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 $s !in! (0,1)$, $0!<!alpha!<!2s!<!n$, $2s!<!m!<!n$, $bar{theta}=max { frac{2}{2^*_{s}(alpha)}, 1-frac{alpha}{s}cdotfrac{1}{2^*_{s}(alpha)}, frac{2^*_{s}(alpha)-frac{alpha}{s}}{2^*_{s}(alpha)-frac{2alpha}{m}} }$, $r=frac{2alpha}{ 2^*_{s}(alpha) }$ 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.