Existence of solution of sub-elliptic equations on the Heisenberg group with critical growth and double singularities

• Autorzy: Chen J. , Rocha E. M.
• Języki publikacji: EN

Abstrakty

EN

For a class of sub-elliptic equations on Heisenberg group HN with Hardy type singularity and critical nonlinear growth, we prove the existence of least energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math.118 (2005), 35–252].

Słowa kluczowe

EN

sub-elliptic equations; Heisenberg group; Least energy solutions

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2013
• Tom: Vol. 33, no. 2
• Strony: 237--254
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Chen J.

• University of Aveiro Department of Mathematics and CIDMA-Center for Research and Development in Mathematics and Applications 3810-193 Aveiro, Portugal

autor

Rocha E. M.

• University of Aveiro Department of Mathematics and CIDMA-Center for Research and Development in Mathematics and Applications 3810-193 Aveiro, Portugal

Bibliografia

• [1] B. Abdellaoui, V. Felli, I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian, Boll. Unione Mat. Ital. 8 (2006), 445-484.
• [2] R.B. Assuncao, P.C. Carriao, O.H. Miyagaki, Critical singular problems via concentration-compactness lemma, J. Math. Anal. Appl. 326 (2007), 137-154.
• [3] J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis, Pure and Applied Mathematics, Wiley Interscience Publications, 1984.
• [4] H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490.
• [5] I. Birindelli, I. Capuzzo Dolcetta, A. Cutri, Indefinite semi-linear equations on the Heisenberg group: a priori bounds and existence, Comm. Partial Differential Equations 23 (1998), 1123-1157.
• [6] I. Birindelli, A. Cutri, A semi-linear problem for the Heisenberg Laplacian, Rend. Semin. Mat. Univ. Padova 94 (1995), 137-153.
• [7] F. Catrina, D.G. Costa, Sharp weighted-norm inequalities for functions with compact support in WLN\{0}, J. Differential Equations 246 (2009), 164-182.
• [8] F. Catrina, Z.Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), 229-258.
• [9] G. Citti, Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian, Ann. Mat. Pura Appl. CLXIX (1995), 375-392.
• [10] G. Citti, F. Uguzzoni, Critical semilinear equations on the Heisenberg group: the effect of topology of the domain, Nonlinear Anal. 46 (2001), 399-417.
• [11] N. Garofalo, D. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Caratheodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. XLIX (1996), 1081-1144.
• [12] Y. Han, P.C. Niu, Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235-252.
• [13] P.C. Niu, H.Q. Zhang, Y. Wang, Hardy type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc. 129 (2001), 3623-3630.
• [14] T. Horiuchi, Best constant in weighted Sobolev inequality with weights being powers of distance from the origin, J. Inequal. Appl. 1 (1997), 275-292.
• [15] D.S. Jerison, J.M. Lee, Extremals of the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1-13.
• [16] P.L. Lions, The concentration compactness principle in the calculus of variations, The limit case, I, II, Rev. Mat. Iberoam. 1 (1985), 145-201, 45-121.
• [17] H. Mokrani, Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential, Comm. Pure Appl. Anal. 8 (2009), 1619-1636.
• [18] A. Szulkin, S. Waliullah, Sign-changing and symmetry-breaking solutions to singular problems, Complex Var. Elliptic Equ. 57 (2012), 1191-1208.
• [19] A. Szulkin, T. Weth, The method of Nehari manifold, [in:] Handbook of Nonconvex Analysis and Applications, D.Y. Gao and D. Motreanu, eds., International Press, Boston (2010), 597-632.
• [20] J.G. Tan, J.F. Yang, On the singular variational problems, Acta Math. Sci. (Ser. B) 24 (2004), 672-690.
• [21] G. Tarantello, On nonhomogeneous elliptic equations involving critical sobolev exponent, Ann. Inst. H. Poincare Anal. Non Lineaire 9 (1992), 281-304.
• [22] S. Terracini, On positive entire solutions to a class of equations with singular coefficient and critical exponent, Adv. Differential Equations 1 (1996), 241-264.
• [23] F. Uguzzoni, A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on half spaces of the Heisenberg group, NoDEA Nonlinear Differential Equations Appl. 6 (1999), 191-206.