Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Subelliptic and parametric equations on Carnot groups

Authors: Giovanni Molica Bisci and Massimiliano Ferrara
Journal: Proc. Amer. Math. Soc. 144 (2016), 3035-3045
MSC (2010): Primary 35J65; Secondary 22E25
Published electronically: November 6, 2015
MathSciNet review: 3487234
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article concerns a class of elliptic equations on Carnot groups depending on one real parameter. Our approach is based on variational methods. More precisely, we establish the existence of at least two weak solutions for the treated problem by using a direct consequence of the celebrated Pucci-Serrin theorem and of a local minimum result for differentiable functionals due to Ricceri.

References [Enhancements On Off] (What's this?)

  • [1] Giovanni Anello, A note on a problem by Ricceri on the Ambrosetti-Rabinowitz condition, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1875-1879 (electronic). MR 2286099 (2007k:35121),
  • [2] Zoltán M. Balogh and Alexandru Kristály, Lions-type compactness and Rubik actions on the Heisenberg group, Calc. Var. Partial Differential Equations 48 (2013), no. 1-2, 89-109. MR 3090536,
  • [3] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2363343 (2009m:22012)
  • [4] Sara Bordoni, Roberta Filippucci, and Patrizia Pucci, Nonlinear elliptic inequalities with gradient terms on the Heisenberg group, Nonlinear Anal. 121 (2015), 262-279. MR 3348926,
  • [5] Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382 (85a:46001)
  • [6] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161-207. MR 0494315 (58 #13215)
  • [7] G. B. Folland and E. M. Stein, Estimates for the $ \bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522. MR 0367477 (51 #3719)
  • [8] Nicola Garofalo and Ermanno Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41 (1992), no. 1, 71-98. MR 1160903 (93h:35071),
  • [9] Alexandru Kristály, Vicenţiu D. Rădulescu, and Csaba György Varga, Variational principles in mathematical physics, geometry, and economics: Qualitative analysis of nonlinear equations and unilateral problems, Encyclopedia of Mathematics and its Applications, vol. 136, Cambridge University Press, Cambridge, 2010. With a foreword by Jean Mawhin. MR 2683404 (2011i:49003)
  • [10] Annunziata Loiudice, Semilinear subelliptic problems with critical growth on Carnot groups, Manuscripta Math. 124 (2007), no. 2, 247-259. MR 2341788 (2008k:35066),
  • [11] Giovanni Molica Bisci and Vicenţiu D. Rădulescu, A characterization for elliptic problems on fractal sets, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2959-2968. MR 3336620,
  • [12] Andrea Pinamonti and Enrico Valdinoci, A Lewy-Stampacchia estimate for variational inequalities in the Heisenberg group, Rend. Istit. Mat. Univ. Trieste 45 (2013), 23-45. MR 3168296
  • [13] Patrizia Pucci and James Serrin, A mountain pass theorem, J. Differential Equations 60 (1985), no. 1, 142-149. MR 808262 (86m:58038),
  • [14] Biagio Ricceri, On a classical existence theorem for nonlinear elliptic equations, Constructive, experimental, and nonlinear analysis (Limoges, 1999), CMS Conf. Proc., vol. 27, Amer. Math. Soc., Providence, RI, 2000, pp. 275-278. MR 1777629 (2001h:35068)
  • [15] Biagio Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), no. 1-2, 401-410. Fixed point theory with applications in nonlinear analysis. MR 1735837 (2001h:47114),
  • [16] Biagio Ricceri, Nonlinear eigenvalue problems, Handbook of nonconvex analysis and applications, Int. Press, Somerville, MA, 2010, pp. 543-595. MR 2768819 (2012h:47139)
  • [17] Biagio Ricceri, A new existence and localization theorem for the Dirichlet problem, Dynam. Systems Appl. 22 (2013), no. 2-3, 317-324. MR 3100206

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35J65, 22E25

Retrieve articles in all journals with MSC (2010): 35J65, 22E25

Additional Information

Giovanni Molica Bisci
Affiliation: Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di Vito, 89100 Reggio, Calabria, Italy

Massimiliano Ferrara
Affiliation: University of Reggio Calabria and CRIOS University Bocconi of Milan, Via dei Bianchi presso Palazzo Zani, 89127 Reggio, Calabria, Italy

Keywords: Subelliptic equations, Carnot groups, multiple solutions, critical point results
Received by editor(s): May 10, 2015
Received by editor(s) in revised form: August 17, 2015, and September 3, 2015
Published electronically: November 6, 2015
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society