Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms

In this paper, we prove two new improved Sobolev inequalities involving weighted Morrey norms in $ {\dot{H}}^s( \mathbb{R}^{n})\times {\dot{H}}^s( \mathbb{R}^{n}) $ and $ {D}^{1, p}( \mathbb{R}^{n})\times{D}^{1, p}( \mathbb{R}^{n}) $. For instance, the corresponding inequality in $ {\dot{H}}^s( \mathbb{R}^{n})\!\times\! {\dot{H}}^s( \mathbb{R}^{n}) $ states that: there exists $ C\! = \!C(n, s, \alpha, \eta_1, \eta_2)\!>\!0 $ such that for each $ (u, v) \!\in\! {\dot{H}}^s( \mathbb{R}^{n})\!\times\! {\dot{H}}^s( \mathbb{R}^{n}) $, $ p\!\in\![2, 2^*_{s}(\alpha)) $ and $ \theta \!\in\! (\bar{\theta}, \frac{2\eta_1}{2^*_{s}(\alpha)}) $, it holds that

$ \Big( \int_{ \mathbb{R}^{n} } \frac{ |u|^{\eta_1} |v|^{\eta_2} } { |y|^{\alpha} } dy \Big)^{ \frac{1}{ 2^*_{s} (\alpha) }} \nonumber \\ \!\leq\! C ||u||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}} ||v||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}+\frac{\eta_2-\eta_1}{2^*_{s} (\alpha)}} ||(uv)||^{\frac{\eta_1}{2^*_{s} (\alpha)}-\frac{\theta}{2}}_{ L^{\frac{p}{2}, \frac{p}{2}(n-2s+r)}(\mathbb{R}^{n}, |y|^{-\frac{p}{2}r}) }, ~~~~(0)$

where $ s \!\in\! (0, 1) $, $ 0\!<\!\alpha\!<\!2s\!<\!n $, $ \eta_1\!+\!\eta_2\! = \!2^*_{s}(\alpha)\!: = \!\frac{2(n-\alpha)}{n-2s} $, $ 1\!<\!\eta_1\!\leq\!\eta_2\!<\!\eta_1\!+\!\frac{\alpha}{s} $, $ \bar{\theta}\! = \!\max \Big\{ \frac{2}{2^*_{s}(\alpha)}, \frac{2\eta_1}{2^*_{s}(\alpha)} -\frac{2t(\frac{\alpha}{2s}-\frac{\alpha}{n})}{2^*_{s}(\alpha) -\frac{2\alpha}{n}}\Big\} $, $ t\! = \!1\!-\!\frac{(\eta_2-\eta_1)s}{\alpha} $ and $ r\! = \!\frac{2\alpha}{ 2^*_{s}(\alpha) } $. This inequality, together with its counterpart in $ {D}^{1, p}( \mathbb{R}^{n})\!\times\!{D}^{1, p}( \mathbb{R}^{n}) $ extend similar Sobolev inequality in $ {\dot{H}}^s( \mathbb{R}^{n}) $ as well as in $ {D}^{1, p}( \mathbb{R}^{n}) $ obtained by G. Palatucci and A. Pisante [Calc. Var., 50 (2014)] to the product spaces $ {\dot{H}}^s( \mathbb{R}^{n})\times {\dot{H}}^s( \mathbb{R}^{n}) $ and $ {D}^{1, p}( \mathbb{R}^{n})\times{D}^{1, p}( \mathbb{R}^{n}) $, respectively.

With the help of the inequality (1), we succeed in obtaining some new existence results for doubly critical elliptic systems involving fractional Laplacian and Hardy terms.