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  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.
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
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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

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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.  Google Scholar

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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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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. 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.  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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