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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

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