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

  • * Corresponding author: Gongbao Li

    * Corresponding author: Gongbao Li 
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  • 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.

    Mathematics Subject Classification: Primary: 35J50; Secondary: 35A23, 35B33.


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