Abstract
In this paper, existence of solutions is established for critical exponential Kirchhoff systems on the Heisenberg group by using the variational method. The novelty of our paper is that not only the nonlinear term has critical exponential growth, but also that Kirchhoff function covers the degenerate case. Moreover, our result is new even for the Euclidean case.
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Alves, C., Boudjeriou, T.: Existence of solution for a class of nonlocal problem via dynamical methods. Rend. Circ. Mat. Palermo 71, 611–632 (2022)
Alves, C.O., Corrês, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)
Ambrosio, V., Isernia, T., Rădulescu, V.D.: Concentration of positive solutions for a class of fractional \(p\)-Kirchhoff type equations. Proc. R. Soc. Edinb. Sect. A 151, 601–651 (2021)
Aouaoui, S.: Multiplicity result for some Kirchhoff-type equations involving exponential growth condition in \({\mathbb{R} }^2\). Commun. Pure Appl. Anal. 15, 1351–1370 (2016)
Bony, J.M.: Principe du Maximum, Inégalité de Harnack et unicité du probléme de Cauchy pour les operateurs elliptiques dégénérés. Ann. Inst. Four. (Grenoble) 19, 277–304 (1969)
Caponi, M., Pucci, P.: Existence theorems for entire solutions of stationary Kirchhoff fractional \(p\)-Laplacian equations. Ann. Mat. Pura Appl. 195, 2099–2129 (2016)
Cohn, W.S., Lu, G.Z.: Best constants for Moser–Trudinger inequalities, fundamental solutions and one-parameter representation formulas on groups of Heisenberg type. Acta Math. Sin. 18, 375–390 (2002)
de Albuquerque, J.C., do Ó, J.M., dos Santos, E.O., Severo, U.B.: On solutions for a class of Kirchhoff systems involving critical growth in \({\mathbb{R}}^2\). Asymptot. Anal. 122(1–2), 69–85 (2021)
Deng, S., Tian, X.: Existence of solutions for Kirchhoff type systems involving \(Q\)-Laplacian operator in Heisenberg group. J. Math. Anal. Appl. 495(1), 124727 (2021)
Figueiredo, G.M., Severo, U.B.: Ground state solution for a Kirchhoff problem with exponential critical growth. Milan J. Math. 84, 23–39 (2016)
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
Leonardi, G.P., Masnou, S.: On the isoperimetric problem in the Heisenberg group \({\mathbb{H} }^n\). Ann. Mat. Pura Appl. 184, 533–553 (2005)
Liang, S., Pucci, P.: Multiple solutions for critical Kirchhoff–Poisson systems in the Heisenberg group. Appl. Math. Lett. 127, 107846 (2022)
Mingqi, X., Molica Bisci, G., Tian, G., Zhang, B.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional \(p\)-Laplacian. Nonlinearity 29, 357–374 (2016)
Mingqi, X., Rădulescu, V.D., Zhang, B.: Nonlocal Kirchhoff problems with singular exponential nonlinearity. Appl. Math. Optim. 84, 915–954 (2021)
Moser, J.: Sharp form of an inequatity by N. Trudinger, Indiana Univer. Math. J. 20(11), 1077–1092 (1971)
Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Nonlinear Analysis—Theory and Methods. Springer Monographs in Mathematics. Springer, Cham (2019)
Pucci, P.: Critical Schrödinger–Hardy systems in the Heisenberg group. Discrete Contin. Dyn. Syst. Ser. S. 12, 375–400 (2019)
Pucci, P.: Existence and multiplicity results for quasilinear elliptic equations in the Heisenberg group. Opuscula Math. 39, 247–257 (2019)
Pucci, P., Temperini, L.: Concentration-compactness results for systems in the Heisenberg group. Opuscula Math. 40, 151–162 (2020)
Pucci, P., Temperini, L.: Existence for \((p, q)\) critical systems in the Heisenberg group. Adv. Nonlinear Anal. 9, 895–922 (2020)
Pucci, P., Temperini, L.: Existence for singular critical exponential \((p, Q)\) equations in the Heisenberg group. Adv. Calc. Var. 15(3), 601–617 (2022)
Pucci, P., Xiang, M., Zhang, B.: Existence and multiplicity of entire solutions for fractional \(p\)-Kirchhoff equations. Adv. Nonlinear Anal. 5, 27–55 (2016)
Trudinger, N.S.: On the imbedding into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967)
Acknowledgements
Li was supported by the Graduate Scientific Research Project of Changchun Normal University (SGSRPCNU [2022], Grant No. 059). Liang was supported by the Foundation for China Postdoctoral Science Foundation (Grant No. 2019M662220), the Research Foundation of Department of Education of Jilin Province (Grant No. JJKH20211161KJ), and the Natural Science Foundation of Jilin Province (Grant no. YDZJ202201ZYTS582). Repovš was supported by the Slovenian Research Agency Program No. P1-0292 and Grants Nos. N1-0278, N1-0114, and N1-0083. The authors thank the anonymous referees for their suggestions and comments.
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Li, S., Liang, S. & Repovš, D.D. On critical exponential Kirchhoff systems on the Heisenberg group. Rend. Circ. Mat. Palermo, II. Ser 72, 2565–2577 (2023). https://doi.org/10.1007/s12215-022-00815-x
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DOI: https://doi.org/10.1007/s12215-022-00815-x