American Institute of Mathematical Sciences

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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  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}) }, ~~~~(1)$
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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020469
References:
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A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

R. B. Assuncao, J. C. Silva and O. H. Miyagaki, A fractional p-Laplacian problem with mul-tiple critical Hardy-Sobolev nonlinearities, Milan J. Math., 88 (2020), 65–97. arXiv: 1906.07227. doi: 10.1007/s00032-020-00308-5.  Google Scholar

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J. Bellazzini, M. Ghimenti, C. Mercuri, V. Moroz and J. V. Schaftingen, Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces, Trans. Amer. Math. Soc., 370 (2018), 8285-8310.  doi: 10.1090/tran/7426.  Google Scholar

[4]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[5]

L. Caffarelli, Non-local diffusions, drifts and games, in: Nonlinear Partial Differential Equations, Abel Symposia No. 7, Springer, Heidelberg, 2012, 37–52. doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and simmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

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Z. Chen and W. Zou, Existence and symmetry of positive ground states for a doubly critical Schrödinger system, Trans. Amer. Math. Soc., 367 (2015), 3599-3646.  doi: 10.1090/S0002-9947-2014-06237-5.  Google Scholar

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J.-L. Chern and C.-S. Lin, Minimizers of Caffarelli-Kohn-Nirenberg inequalities with the singularity on the boundary, Arch. Ration. Mech. Anal., 197 (2010), 401-432.  doi: 10.1007/s00205-009-0269-y.  Google Scholar

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G. Difazio and M. A. Ragusa, Interior Estimates in Morrey Spaces for Strong Solutions to Nondivergence Form Equations with Discontinuous Coefficients, J. Funct. Anal., 112 (1993), 241-256.  doi: 10.1006/jfan.1993.1032.  Google Scholar

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S. Dipierro, L. Montoro, I. Peral and B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 99, 29 pp. doi: 10.1007/s00526-016-1032-5.  Google Scholar

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R. Filippucci, P. Pucci and F. Robert, On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177.  doi: 10.1016/j.matpur.2008.09.008.  Google Scholar

[14]

R. L. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc., 21 (2008), 925-950.  doi: 10.1090/S0894-0347-07-00582-6.  Google Scholar

[15]

N. Ghoussoub and S. Shakerian, Borderline variational problems involving fractional Laplacians and critical singularities, Adv. Nonlinear Stud., 15 (2015), 527-555.  doi: 10.1515/ans-2015-0302.  Google Scholar

[16]

N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Mathematical Surveys and Monographs, vol. 187, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/surv/187.  Google Scholar

[17]

N. Ghoussoub and F. Robert, The Hardy-Schr$\ddot{o}$dinger operator with interior singularity: The remaining cases, Calc. Var. Partial Differ. Equ., 56 (2017). doi: 10.1007/s00526-017-1238-1.  Google Scholar

[18]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.  doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar

[19]

T. Gou and L. Jeanjean, Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity, 31 (2018), 2319-2345.  doi: 10.1088/1361-6544/aab0bf.  Google Scholar

[20]

Y. Huang and D. Kang, On the singular elliptic systems involving multiple critical Sobolev exponents, Nonlinear Anal., 74 (2011), 400-412.  doi: 10.1016/j.na.2010.08.051.  Google Scholar

[21]

D. Kang and G. Li, On the elliptic problems involving multi-singular inverse square potentials and multi-critical Sobolev-Hardy exponents, Nonlinear Anal., 66 (2007), 1806-1816.  doi: 10.1016/j.na.2006.02.026.  Google Scholar

[22]

A. E. Khalil, S. Kellati and A. Touzani, On the principal frequency curve of the p-biharmonic operator, Arab J. Math. Sci., 17 (2011), 89-99.  doi: 10.1016/j.ajmsc.2011.01.002.  Google Scholar

[23]

Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr., 282 (2009), 219-231.  doi: 10.1002/mana.200610733.  Google Scholar

[24]

G. Li and T. Yang, The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${ \mathbb{R}}^n$ with a Hardy term, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 1808–1830. arXiv: 1908.02536. doi: 10.1007/s10473-020-0613-8.  Google Scholar

[25]

E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Volume 14 of Graduate Studies in Mathematics, Amer. Math. Soc., 2001. doi: 10.1090/gsm/014.  Google Scholar

[26]

L. D'Ambrosio and E. Jannelli, Nonlinear critical problems for the biharmonic operator with Hardy potential, Calc. Var. Partial Differ. Equ., 54 (2015), 365-396.  doi: 10.1007/s00526-014-0789-7.  Google Scholar

[27]

A. L. Mazzucato, Besov-Morrey spaces: Function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 355 (2003), 1297-1364.  doi: 10.1090/S0002-9947-02-03214-2.  Google Scholar

[28]

C. Mercuri, V. Moroz and J. V. Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency, Calc. Var. Partial Differ. Equ., 55 (2016). doi: 10.1007/s00526-016-1079-3.  Google Scholar

[29]

C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.  doi: 10.1090/S0002-9947-1938-1501936-8.  Google Scholar

[30]

B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274.  doi: 10.1090/S0002-9947-1974-0340523-6.  Google Scholar

[31]

E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[32]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.  Google Scholar

[33]

S. Rastegarzadeh, N. Nyamoradi and V. Ambrosio, Existence and multiplicity of solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities, J. Fixed Point Theory Appl., 21 (2019), 1-22.  doi: 10.1007/s11784-018-0653-z.  Google Scholar

[34]

Y. Sawano, Generalized Morrey spaces for non-doubling measures, Nonlinear Differ. Equ. Appl., 15 (2008), 413-425.  doi: 10.1007/s00030-008-6032-5.  Google Scholar

[35]

E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math., 114 (1992), 813-874.  doi: 10.2307/2374799.  Google Scholar

[36]

R. Servadei and E. Raffaella, Variational methods for non-local operators of eliliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[37]

Y. Su, H. Chen, S. Liu and G. Che, Ground state solution of p-Laplacian equation with finite many critical nonlinearities, Complex Var. Elliptic Equ., 65 (2020). doi: 10.1080/17476933.2020.1720005.  Google Scholar

[38]

Y. Wang and Y. Shen, Nonlinear biharmonic equations with Hardy potential and critical parameter, J. Math. Anal. Appl., 355 (2009), 649-660.  doi: 10.1016/j.jmaa.2009.01.076.  Google Scholar

[39]

J. Wang and J. Shi, Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction, Calc. Var. Partial Differ. Equ., 56 (2017). doi: 10.1007/s00526-017-1268-8.  Google Scholar

[40]

L. Wang, B. Zhang and H. Zhang, Fractional Laplacian system involving doubly critical nonlinearities in $ \mathbb{R}^N$, Electron. J. Qual. Theory Differ. Equ., 57 (2017), 1-17.  doi: 10.14232/ejqtde.2017.1.57.  Google Scholar

[41]

J. Yang, Fractional Hardy-Sobolev inequality in $ \mathbb{R}^N$, Nonlinear Anal., 119 (2015), 179-185.  doi: 10.1016/j.na.2014.09.009.  Google Scholar

[42]

J. Yang and F. Wu, Doubly critical problems involving fractional Laplacians in ${ \mathbb{R}}^N$, Adv. Nonlinear Stud., 17 (2017), 677-690.  doi: 10.1515/ans-2016-6012.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

R. B. Assuncao, J. C. Silva and O. H. Miyagaki, A fractional p-Laplacian problem with mul-tiple critical Hardy-Sobolev nonlinearities, Milan J. Math., 88 (2020), 65–97. arXiv: 1906.07227. doi: 10.1007/s00032-020-00308-5.  Google Scholar

[3]

J. Bellazzini, M. Ghimenti, C. Mercuri, V. Moroz and J. V. Schaftingen, Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces, Trans. Amer. Math. Soc., 370 (2018), 8285-8310.  doi: 10.1090/tran/7426.  Google Scholar

[4]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[5]

L. Caffarelli, Non-local diffusions, drifts and games, in: Nonlinear Partial Differential Equations, Abel Symposia No. 7, Springer, Heidelberg, 2012, 37–52. doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[7]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and simmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[8]

W. Chen, Fractional elliptic problems with two critical Sobolev-Hardy exponents, Electron. J. Differential Equations, 2018 (2018).  Google Scholar

[9]

Z. Chen and W. Zou, Existence and symmetry of positive ground states for a doubly critical Schrödinger system, Trans. Amer. Math. Soc., 367 (2015), 3599-3646.  doi: 10.1090/S0002-9947-2014-06237-5.  Google Scholar

[10]

J.-L. Chern and C.-S. Lin, Minimizers of Caffarelli-Kohn-Nirenberg inequalities with the singularity on the boundary, Arch. Ration. Mech. Anal., 197 (2010), 401-432.  doi: 10.1007/s00205-009-0269-y.  Google Scholar

[11]

G. Difazio and M. A. Ragusa, Interior Estimates in Morrey Spaces for Strong Solutions to Nondivergence Form Equations with Discontinuous Coefficients, J. Funct. Anal., 112 (1993), 241-256.  doi: 10.1006/jfan.1993.1032.  Google Scholar

[12]

S. Dipierro, L. Montoro, I. Peral and B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 99, 29 pp. doi: 10.1007/s00526-016-1032-5.  Google Scholar

[13]

R. Filippucci, P. Pucci and F. Robert, On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177.  doi: 10.1016/j.matpur.2008.09.008.  Google Scholar

[14]

R. L. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc., 21 (2008), 925-950.  doi: 10.1090/S0894-0347-07-00582-6.  Google Scholar

[15]

N. Ghoussoub and S. Shakerian, Borderline variational problems involving fractional Laplacians and critical singularities, Adv. Nonlinear Stud., 15 (2015), 527-555.  doi: 10.1515/ans-2015-0302.  Google Scholar

[16]

N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Mathematical Surveys and Monographs, vol. 187, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/surv/187.  Google Scholar

[17]

N. Ghoussoub and F. Robert, The Hardy-Schr$\ddot{o}$dinger operator with interior singularity: The remaining cases, Calc. Var. Partial Differ. Equ., 56 (2017). doi: 10.1007/s00526-017-1238-1.  Google Scholar

[18]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.  doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar

[19]

T. Gou and L. Jeanjean, Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity, 31 (2018), 2319-2345.  doi: 10.1088/1361-6544/aab0bf.  Google Scholar

[20]

Y. Huang and D. Kang, On the singular elliptic systems involving multiple critical Sobolev exponents, Nonlinear Anal., 74 (2011), 400-412.  doi: 10.1016/j.na.2010.08.051.  Google Scholar

[21]

D. Kang and G. Li, On the elliptic problems involving multi-singular inverse square potentials and multi-critical Sobolev-Hardy exponents, Nonlinear Anal., 66 (2007), 1806-1816.  doi: 10.1016/j.na.2006.02.026.  Google Scholar

[22]

A. E. Khalil, S. Kellati and A. Touzani, On the principal frequency curve of the p-biharmonic operator, Arab J. Math. Sci., 17 (2011), 89-99.  doi: 10.1016/j.ajmsc.2011.01.002.  Google Scholar

[23]

Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr., 282 (2009), 219-231.  doi: 10.1002/mana.200610733.  Google Scholar

[24]

G. Li and T. Yang, The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${ \mathbb{R}}^n$ with a Hardy term, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 1808–1830. arXiv: 1908.02536. doi: 10.1007/s10473-020-0613-8.  Google Scholar

[25]

E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Volume 14 of Graduate Studies in Mathematics, Amer. Math. Soc., 2001. doi: 10.1090/gsm/014.  Google Scholar

[26]

L. D'Ambrosio and E. Jannelli, Nonlinear critical problems for the biharmonic operator with Hardy potential, Calc. Var. Partial Differ. Equ., 54 (2015), 365-396.  doi: 10.1007/s00526-014-0789-7.  Google Scholar

[27]

A. L. Mazzucato, Besov-Morrey spaces: Function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 355 (2003), 1297-1364.  doi: 10.1090/S0002-9947-02-03214-2.  Google Scholar

[28]

C. Mercuri, V. Moroz and J. V. Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency, Calc. Var. Partial Differ. Equ., 55 (2016). doi: 10.1007/s00526-016-1079-3.  Google Scholar

[29]

C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.  doi: 10.1090/S0002-9947-1938-1501936-8.  Google Scholar

[30]

B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274.  doi: 10.1090/S0002-9947-1974-0340523-6.  Google Scholar

[31]

E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[32]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.  Google Scholar

[33]

S. Rastegarzadeh, N. Nyamoradi and V. Ambrosio, Existence and multiplicity of solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities, J. Fixed Point Theory Appl., 21 (2019), 1-22.  doi: 10.1007/s11784-018-0653-z.  Google Scholar

[34]

Y. Sawano, Generalized Morrey spaces for non-doubling measures, Nonlinear Differ. Equ. Appl., 15 (2008), 413-425.  doi: 10.1007/s00030-008-6032-5.  Google Scholar

[35]

E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math., 114 (1992), 813-874.  doi: 10.2307/2374799.  Google Scholar

[36]

R. Servadei and E. Raffaella, Variational methods for non-local operators of eliliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[37]

Y. Su, H. Chen, S. Liu and G. Che, Ground state solution of p-Laplacian equation with finite many critical nonlinearities, Complex Var. Elliptic Equ., 65 (2020). doi: 10.1080/17476933.2020.1720005.  Google Scholar

[38]

Y. Wang and Y. Shen, Nonlinear biharmonic equations with Hardy potential and critical parameter, J. Math. Anal. Appl., 355 (2009), 649-660.  doi: 10.1016/j.jmaa.2009.01.076.  Google Scholar

[39]

J. Wang and J. Shi, Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction, Calc. Var. Partial Differ. Equ., 56 (2017). doi: 10.1007/s00526-017-1268-8.  Google Scholar

[40]

L. Wang, B. Zhang and H. Zhang, Fractional Laplacian system involving doubly critical nonlinearities in $ \mathbb{R}^N$, Electron. J. Qual. Theory Differ. Equ., 57 (2017), 1-17.  doi: 10.14232/ejqtde.2017.1.57.  Google Scholar

[41]

J. Yang, Fractional Hardy-Sobolev inequality in $ \mathbb{R}^N$, Nonlinear Anal., 119 (2015), 179-185.  doi: 10.1016/j.na.2014.09.009.  Google Scholar

[42]

J. Yang and F. Wu, Doubly critical problems involving fractional Laplacians in ${ \mathbb{R}}^N$, Adv. Nonlinear Stud., 17 (2017), 677-690.  doi: 10.1515/ans-2016-6012.  Google Scholar

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