April  2019, 12(2): 375-400. doi: 10.3934/dcdss.2019025

Critical Schrödinger-Hardy systems in the Heisenberg group

Department of Mathematics and Informatics, University of Perugia, Via Vanvitelli, 1, 06123 Perugia, Italy

Dedicated to Professor Vicentiu D. Radulescu on the occasion of his 60th birthday, with high feelings of admiration for his notable contributions in Mathematics and great affection

Received  May 2017 Revised  December 2017 Published  August 2018

The paper is focused on existence of nontrivial solutions of a Schrödinger-Hardy system in the Heisenberg group, involving critical nonlinearities. Existence is obtained by an application of the mountain pass theorem and the Ekeland variational principle, but there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the Hardy terms as well as critical nonlinearities.

Citation: Patrizia Pucci. Critical Schrödinger-Hardy systems in the Heisenberg group. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 375-400. doi: 10.3934/dcdss.2019025
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, second eds., Academic Press, New York–London, 2003.

[2]

G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 977-1009.  doi: 10.1007/s00030-012-0193-y.

[3]

Z. M. Balogh and A. Kristály, Lions-type compactness and Rubik actions on the Heisenberg group, Calc. Var. Partial Differential Equations, 48 (1995), 89-109.  doi: 10.1007/s00526-012-0543-y.

[4]

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.

[5]

S. Bordoni and P. Pucci, Schrödinger–Hardy systems involving two Laplacian operators in the Heisenberg group, Bull. Sci. Math., 146 (2018), 50-88. doi: 10.1016/j.bulsci.2018.03.001.

[6]

M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x.

[7]

C. Chen, Infinitely many solutions to a class of quasilinear Schrödinger system in $\mathbb{R}^N$, Appl. Math. Lett., 52 (2016), 176-182.  doi: 10.1016/j.aml.2015.09.007.

[8]

W. Chen and M. Squassina, Critical nonlocal systems with concave-convex powers, Adv. Nonlinear Stud., 16 (2016), 176-182.  doi: 10.1515/ans-2015-5055.

[9]

J. Y. Chu, Z. W. Wei and Q. Y. Wu, Lp and BMO bounds for weighted Hardy operators on the Heisenberg group J. Inequal. Appl., (2016), Paper No. 282, 12 pp. doi: 10.1186/s13660-016-1222-x.

[10]

L. D'Ambrosio, Hardy-type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 4 (2005), 451-486. 

[11]

F. Demengel and E. Hebey, On some nonlinear equations on compact Riemannian manifolds, Adv. Differential Equations, 3 (1998), 533-574. 

[12]

A. FiscellaP. Pucci and S. Saldi, Existence of entire solutions for Schrödinger-Hardy systems involving two fractional operators, Nonlinear Anal., 158 (2017), 109-131.  doi: 10.1016/j.na.2017.04.005.

[13]

A. Fiscella, P. Pucci and B. Zhang, p–fractional Hardy–Schrödinger–Kirchhoff Systems with Critical Nonlinearities, submitted for publication, pages 22.

[14]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Math., 13 (1975), 161-207.  doi: 10.1007/BF02386204.

[15]

G. B. Folland and E. M. Stein, Estimates for the b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522.  doi: 10.1002/cpa.3160270403.

[16]

B. FranchiC. Gutierrez and R. L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. PDE, 19 (1994), 523-604.  doi: 10.1080/03605309408821025.

[17]

Y. FuH. Li and P. Pucci, Existence of nonnegative solutions for a class of systems involving fractional (p, q)-Laplacian operators, Chin. Ann. Math. Ser. B, 39 (2018), 357-372.  doi: 10.1007/s11401-018-1069-1.

[18]

N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier, 40 (1990), 313-356.  doi: 10.5802/aif.1215.

[19]

N. Garofalo and D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math., 49 (1996), 1081-1144.  doi: 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A.

[20]

P. Han, The effect of the domian topology on the number of positive solutions of an elliptic system involving critical Sobolev exponents, Houston J. Math., 32 (2006), 1241-1257. 

[21]

L. Hőrmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.

[22]

S. P. Ivanov, D. N. Vassilev, Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, ⅹⅷ+219 pp., 2011. doi: 10.1142/9789814295710.

[23]

G. P. Leonardi and S. Masnou, On the isoperimetric problem in the Heisenberg group $\mathbb H^n$, Ann. Mat. Pura Appl.(4), 184 (2005), 533-553.  doi: 10.1007/s10231-004-0127-3.

[24]

A. Loiudice, Improved Sobolev inequalities on the Heisenberg group, Nonlinear Anal., 62 (2005), 953-962.  doi: 10.1016/j.na.2005.04.015.

[25]

M. MagliaroL. MariP. Mastrolia and M. Rigoli, Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group, J. Diff. Equations, 250 (2011), 2643-2670.  doi: 10.1016/j.jde.2011.01.006.

[26]

G. MingioneA. Zatorska-Goldstein and X. Zhong, Gradient regularity for elliptic equations in the Heisenberg group, Adv. Math., 222 (2009), 62-129.  doi: 10.1016/j.aim.2009.03.016.

[27]

X. Mingqi, V. Radulescu and B. Zhang, Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities, ESAIM Control Optim. Calc. Var., (2017), pages 28. doi: 10.1051/cocv/2017036.

[28]

P. NiuH. Zhang and Y. Wang, Hardy-type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc, 129 (2001), 3623-3630.  doi: 10.1090/S0002-9939-01-06011-7.

[29]

P. PucciM. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in ${\mathbb {R}}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.

[30]

D. Ricciotti, p–Laplace Equation in the Heisenberg Group. Regularity of Solutions, Springer Briefs in Mathematics, BCAM Basque Center for Applied Mathematics, Bilbao, ⅹⅳ+87 pp., 2015. doi: 10.1007/978-3-319-23790-9.

[31]

N. Varopoulos, Analysis on nilpotent Lie groups, J. Funct. Anal., 66 (1986), 406-431.  doi: 10.1016/0022-1236(86)90066-2.

[32]

N. Varopoulos, Sobolev inequalities on Lie groups and symmetric spaces, J. Funct. Anal., 86 (1989), 19-40.  doi: 10.1016/0022-1236(89)90063-3.

[33]

D. Vassilev, Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups, Pacific J. Math., 227 (2006), 361-397.  doi: 10.2140/pjm.2006.227.361.

show all references

Dedicated to Professor Vicentiu D. Radulescu on the occasion of his 60th birthday, with high feelings of admiration for his notable contributions in Mathematics and great affection

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, second eds., Academic Press, New York–London, 2003.

[2]

G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 977-1009.  doi: 10.1007/s00030-012-0193-y.

[3]

Z. M. Balogh and A. Kristály, Lions-type compactness and Rubik actions on the Heisenberg group, Calc. Var. Partial Differential Equations, 48 (1995), 89-109.  doi: 10.1007/s00526-012-0543-y.

[4]

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.

[5]

S. Bordoni and P. Pucci, Schrödinger–Hardy systems involving two Laplacian operators in the Heisenberg group, Bull. Sci. Math., 146 (2018), 50-88. doi: 10.1016/j.bulsci.2018.03.001.

[6]

M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x.

[7]

C. Chen, Infinitely many solutions to a class of quasilinear Schrödinger system in $\mathbb{R}^N$, Appl. Math. Lett., 52 (2016), 176-182.  doi: 10.1016/j.aml.2015.09.007.

[8]

W. Chen and M. Squassina, Critical nonlocal systems with concave-convex powers, Adv. Nonlinear Stud., 16 (2016), 176-182.  doi: 10.1515/ans-2015-5055.

[9]

J. Y. Chu, Z. W. Wei and Q. Y. Wu, Lp and BMO bounds for weighted Hardy operators on the Heisenberg group J. Inequal. Appl., (2016), Paper No. 282, 12 pp. doi: 10.1186/s13660-016-1222-x.

[10]

L. D'Ambrosio, Hardy-type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 4 (2005), 451-486. 

[11]

F. Demengel and E. Hebey, On some nonlinear equations on compact Riemannian manifolds, Adv. Differential Equations, 3 (1998), 533-574. 

[12]

A. FiscellaP. Pucci and S. Saldi, Existence of entire solutions for Schrödinger-Hardy systems involving two fractional operators, Nonlinear Anal., 158 (2017), 109-131.  doi: 10.1016/j.na.2017.04.005.

[13]

A. Fiscella, P. Pucci and B. Zhang, p–fractional Hardy–Schrödinger–Kirchhoff Systems with Critical Nonlinearities, submitted for publication, pages 22.

[14]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Math., 13 (1975), 161-207.  doi: 10.1007/BF02386204.

[15]

G. B. Folland and E. M. Stein, Estimates for the b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522.  doi: 10.1002/cpa.3160270403.

[16]

B. FranchiC. Gutierrez and R. L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. PDE, 19 (1994), 523-604.  doi: 10.1080/03605309408821025.

[17]

Y. FuH. Li and P. Pucci, Existence of nonnegative solutions for a class of systems involving fractional (p, q)-Laplacian operators, Chin. Ann. Math. Ser. B, 39 (2018), 357-372.  doi: 10.1007/s11401-018-1069-1.

[18]

N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier, 40 (1990), 313-356.  doi: 10.5802/aif.1215.

[19]

N. Garofalo and D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math., 49 (1996), 1081-1144.  doi: 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A.

[20]

P. Han, The effect of the domian topology on the number of positive solutions of an elliptic system involving critical Sobolev exponents, Houston J. Math., 32 (2006), 1241-1257. 

[21]

L. Hőrmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.

[22]

S. P. Ivanov, D. N. Vassilev, Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, ⅹⅷ+219 pp., 2011. doi: 10.1142/9789814295710.

[23]

G. P. Leonardi and S. Masnou, On the isoperimetric problem in the Heisenberg group $\mathbb H^n$, Ann. Mat. Pura Appl.(4), 184 (2005), 533-553.  doi: 10.1007/s10231-004-0127-3.

[24]

A. Loiudice, Improved Sobolev inequalities on the Heisenberg group, Nonlinear Anal., 62 (2005), 953-962.  doi: 10.1016/j.na.2005.04.015.

[25]

M. MagliaroL. MariP. Mastrolia and M. Rigoli, Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group, J. Diff. Equations, 250 (2011), 2643-2670.  doi: 10.1016/j.jde.2011.01.006.

[26]

G. MingioneA. Zatorska-Goldstein and X. Zhong, Gradient regularity for elliptic equations in the Heisenberg group, Adv. Math., 222 (2009), 62-129.  doi: 10.1016/j.aim.2009.03.016.

[27]

X. Mingqi, V. Radulescu and B. Zhang, Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities, ESAIM Control Optim. Calc. Var., (2017), pages 28. doi: 10.1051/cocv/2017036.

[28]

P. NiuH. Zhang and Y. Wang, Hardy-type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc, 129 (2001), 3623-3630.  doi: 10.1090/S0002-9939-01-06011-7.

[29]

P. PucciM. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in ${\mathbb {R}}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.

[30]

D. Ricciotti, p–Laplace Equation in the Heisenberg Group. Regularity of Solutions, Springer Briefs in Mathematics, BCAM Basque Center for Applied Mathematics, Bilbao, ⅹⅳ+87 pp., 2015. doi: 10.1007/978-3-319-23790-9.

[31]

N. Varopoulos, Analysis on nilpotent Lie groups, J. Funct. Anal., 66 (1986), 406-431.  doi: 10.1016/0022-1236(86)90066-2.

[32]

N. Varopoulos, Sobolev inequalities on Lie groups and symmetric spaces, J. Funct. Anal., 86 (1989), 19-40.  doi: 10.1016/0022-1236(89)90063-3.

[33]

D. Vassilev, Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups, Pacific J. Math., 227 (2006), 361-397.  doi: 10.2140/pjm.2006.227.361.

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