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

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.
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
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.

[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.

[3]

J. BellazziniM. GhimentiC. MercuriV. 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.

[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.

[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.

[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.

[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.

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

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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.

[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.

[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.

[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.

[13]

R. FilippucciP. 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.

[14]

R. L. FrankE. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

A. E. KhalilS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

E. D. NezzaG. 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.

[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.

[33]

S. RastegarzadehN. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

L. WangB. 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.

[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.

[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.

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.

[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.

[3]

J. BellazziniM. GhimentiC. MercuriV. 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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

R. FilippucciP. 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.

[14]

R. L. FrankE. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

A. E. KhalilS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

E. D. NezzaG. 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.

[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.

[33]

S. RastegarzadehN. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

L. WangB. 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.

[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.

[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.

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