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

Gongbao Li; Tao Yang

doi:10.3934/dcdss.2020469

Previous Article
Solutions to Chern-Simons-Schrödinger systems with external potential
DCDS-S Home
This Issue
Next Article
The Orlicz-Minkowski problem for polytopes

June 2021, 14(6): 1945-1966. doi: 10.3934/dcdss.2020469

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

Gongbao Li ^, and Tao Yang

Hubei Key Laboratory of Mathematical Sciences & School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

^* Corresponding author: Gongbao Li

Received July 2020 Revised September 2020 Published June 2021 Early access November 2020

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.

Keywords: New improved Sobolev inequality, product spaces, weighted Morrey spaces, doubly critical elliptic systems, fractional Laplacian, p-Laplacian, Hardy term.

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

Citation: Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1945-1966. doi: 10.3934/dcdss.2020469