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.
Categories: math.AP