Multiple solutions for a class of nonlinear elliptic equations on Carnot groups

Document Type : Research Paper


Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran


In this paper, using variational methods and critical point theory we establish the existence of multiple solutions for a class of elliptic equations on Carnot groups depending on one real positive parameter and involving a subcritical nonlinearity. Some recent results are extended and improved.


[1] S. Bordoni, R. Filippucci, and P. Pucci, Nonlinear elliptic inequalities with gradient terms on the Heisenberg group, Nonlinear Anal. TMA 121 (2015), 262–279.
[2] M.Z. Balogh and A. Kristaly, Lions-type compactness and Rubik actions on the Heisen- berg group, Calc. Var. Partial Differ. Equ. 48 (2013), 89–109.
[3] G. Caristi, S. Heidarkhani, A. Salari, and S.A. Tersian, Multiple solutions for degenerate nonlocal problems, Appl. Math. Lett. 84 (2018), 26–33.
[4] N.T. Chung and H.Q. Toan, Multiple solutions for a class of degenerate nonlocal problems involving sublinear nonlinearities, Matematiche (Catania) 69 (2014), 171–182.
[5] L. D’Ambrosio and E. Mitidieri, Entire solutions of quasilinear elliptic systems on Carnot groups, Proc. Steklov Inst. Math. 283 (2013), 3–19.
[6] G.B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161–207.
[7] G.B. Folland and E.M. Stein, Estimates for the ∂b complex and analysis on the Heisenberg group, Commun. Pure Appl. Math. 27 (1974), 429–522.
[8] M. Ferrara, G. Molica Bisci and D. Repovs, Nonlinear elliptic equations on Carnot groups, RACSAM 111 (2017), 707–718.
[9] N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41 (1992), 71–98.
[10] A. Loiudice, Semilinear subelliptic problems with critical growth on Carnot groups, Manuscripta Math. 124 (2007), 247–259.
[11] J. Mawhin, Some boundary value problems for Hartman-type perturbations of the ordinary vector p-Laplacian, Nonlinear Anal. TMA 40 (2000), 497–503.
[12] J. J. Manfredi and G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007), 485–544.
[13] G. Molica Bisci and M. Ferrara, Subelliptic and parametric equations on Carnot groups, Proc. Amer. Math. Soc. 144 (2016), no. 7, 3035–3045.
[14] G. Mingione, A. Zatorska-Goldestein and X. Zhong, Gradient regularity for elliptic equations in the Heisenberg group, Adv. Math. 222 (2009), 62–129.
[15] A. Pinamonti and E. Valdinoci, A Lewy-Stampacchia estimate for variational inequalities in the Heisenberg group, Rend. Istit. Mat. Univ. Trieste. 45 (2013), 1–22.
[16] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, Berlin, 1989.
[17] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol., 65, The American Mathematical Society, Providence, RI, Published for the Conference Board of the Mathematical Sciences, Washington, DC., 1986.
[18] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. III. Springer, Berlin, 1985.
[19] D. Zhang, Multiple solutions of nonlinear impulsive differential equations with Dirichlet boundary conditions via variational method, Results. Math. 63 (2013), 611–628.
[20] D. Zhang and B. Dai, Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions, Comput. Math. Appl. 61 (2011), 3153–3160.
[21] Z. Zhang and R. Yuan, An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Anal. RWA 11 (2010), 155–162.
Volume 15, Issue 3
March 2024
Pages 193-199
  • Receive Date: 01 January 2022
  • Revise Date: 03 March 2023
  • Accept Date: 23 March 2023