Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications. (English) Zbl 1278.42033
Authors’ abstract: “Let \(\mathbb{H }= \mathbb{C}^n\times \mathbb{R}\) be the \(n\)-dimensional Heisenberg group, \(Q = 2n + 2\) be the homogeneous dimension of \(\mathbb{H}\), and \(\rho(\xi ) = (|z|^4 + t^2)^{ 1 /4}\) be the homogeneous norm of \(\xi = (z, t)\in \mathbb{H}\). Then we prove a sharp Moser-Trudinger inequality on \(\mathbb{H}\). [...] Our result extends the sharp Moser-Trudinger inequality by W. Cohn and G. Lu [Indiana Univ. Math. J. 50, No. 4, 1567–1591 (2001; Zbl 1019.43009)] on domains of finite measure on \(\mathbb{H}\) and sharpens the recent result of W. S. Cohn et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 12, 4483–4495 (2012; Zbl 1254.35063)]. We carry out a completely different and much simpler argument than that of W. S. Cohn et al. [loc. cit] to conclude the critical case. Our method avoids using the rearrangement argument which is not available in an optimal way on the Heisenberg group and can be used in more general settings such as Riemanian manifolds, appropriate metric spaces, etc. As applications, we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of \(Q\)-Laplacian type on the Heisenberg group [...].”
Reviewer: Yu Liu (Beijing)
MSC:
| 42B37 | Harmonic analysis and PDEs |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J62 | Quasilinear elliptic equations |
| 22E30 | Analysis on real and complex Lie groups |
Keywords:
best constants; Moser-Trudinger inequality; Heisenberg group; Mountain-Pass theorem; subelliptic equations of exponential growth; \(Q\)-sublaplacian; existence and multiplicity of nontrivial solutionsReferences:
| [1] | Adachi, S.; Tanaka, K., Trudinger type inequalities in \(R^N\) and their best exponents, Proc. Amer. Math. Soc., 128, 2051-2057 (1999) · Zbl 0980.46020 |
| [2] | Adams, D. R., A sharp inequality of J. Moser for higher order derivatives, Ann. of Math., 128, 2, 385-398 (1988) · Zbl 0672.31008 |
| [3] | Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the \(n\)-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 17, 3, 393-413 (1990) · Zbl 0732.35028 |
| [4] | Adimurthi; Druet, O., Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality, Comm. Partial Differential Equations, 29, 1-2, 295-322 (2004) · Zbl 1076.46022 |
| [5] | Adimurthi; Struwe, M., Global compactness properties of semilinear elliptic equations with critical exponential growth, J. Funct. Anal., 175, 1, 125-167 (2000) · Zbl 0956.35045 |
| [6] | Adimurthi; Yang, Y., An interpolation of Hardy inequality and Trudinger-Moser inequality in \(R^N\) and its applications, Int. Math. Res. Not. IMRN, 13, 2394-2426 (2010) · Zbl 1198.35012 |
| [7] | Aubin, T., Best constants in the Sobolev imbedding theorem: the Yamabe problem, (Yau, S.-T., Seminar on Differential Geometry (1982), Princeton University: Princeton University Princeton), 173-184 · Zbl 0483.53041 |
| [8] | Balogh, J.; Manfredi, J.; Tyson, J., Fundamental solution for the \(Q\)-Laplacian and sharp Moser-Trudinger inequality in Carnot groups, J. Funct. Anal., 204, 1, 35-49 (2003) · Zbl 1080.22003 |
| [9] | W. Beckner, Embedding estimates and fractional smoothness. arXiv:1206.4215v2; W. Beckner, Embedding estimates and fractional smoothness. arXiv:1206.4215v2 · Zbl 1316.46032 |
| [10] | Beckner, W., Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc., 136, 1871-1885 (2008) · Zbl 1221.42016 |
| [11] | Beckner, W., Weighted inequalities and Stein-Weiss potentials, Forum Math., 20, 4, 587-606 (2008) · Zbl 1149.42006 |
| [12] | Beckner, W., Pitt’s inequality and the fractional Laplacian: sharp error estimates, Forum Math., 24, 177-209 (2012) · Zbl 1244.42005 |
| [13] | Bonfiglioli, B.; Lanconelli, E.; Uguzzoni, F., (Stratified Lie Groups and Potential Theory for their Sub-Laplacians. Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Springer Monographs in Mathematics (2007), Springer: Springer Berlin) · Zbl 1128.43001 |
| [14] | T. Branson, L. Fontana, C. Morpurgo, Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere, Ann. of Math. (in press). arXiv:0712.3905v3; T. Branson, L. Fontana, C. Morpurgo, Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere, Ann. of Math. (in press). arXiv:0712.3905v3 · Zbl 1334.35366 |
| [15] | Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88, 3, 486-490 (1983) · Zbl 0526.46037 |
| [16] | Cao, D. M., Nontrivial solution of semilinear elliptic equation with critical exponent in \(R^2\), Comm. Partial Differential Equations, 17, 3-4, 407-435 (1992) · Zbl 0763.35034 |
| [17] | Carleson, L.; Chang, S.-Y. A., On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2), 110, 2, 113-127 (1986) · Zbl 0619.58013 |
| [18] | Cohn, W. S.; Lam, N.; Lu, G.; Yang, Y., The Moser-Trudinger inequality in unbounded domains of Heisenberg group and sub-elliptic equations, Nonlinear Anal., 75, 12, 4483-4495 (2012) · Zbl 1254.35063 |
| [19] | Cohn, W. S.; Lu, G., Best constants for Moser-Trudinger inequalities on the Heisenberg group, Indiana Univ. Math. J., 50, 4, 1567-1591 (2001) · Zbl 1019.43009 |
| [20] | Cohn, W. S.; Lu, G., Best constants for Moser-Trudinger inequalities, fundamental solutions and one-parameter representation formulas on groups of Heisenberg type, Acta Math. Sin. (Engl. Ser.), 18, 2, 375-390 (2002) · Zbl 1011.22004 |
| [21] | Cohn, W. S.; Lu, G., Sharp constants for Moser-Trudinger inequalities on spheres in complex space \(C^n\), Comm. Pure Appl. Math., 57, 11, 1458-1493 (2004) · Zbl 1063.35060 |
| [22] | de Figueiredo, D. G.; do Ó, J. M.; Ruf, B., On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure Appl. Math., 55, 2, 135-152 (2002) · Zbl 1040.35029 |
| [23] | de Figueiredo, D. G.; Miyagaki, O. H.; Ruf, B., Elliptic equations in \(R^2\) with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3, 2, 139-153 (1995) · Zbl 0820.35060 |
| [24] | do Ó, J. M.; Medeiros, E.; Severo, U., On a quasilinear nonhomogeneous elliptic equation with critical growth in \(R^n\), J. Differential Equations, 246, 4, 1363-1386 (2009) · Zbl 1159.35027 |
| [25] | Duc, D. M.; Thanh Vu, N., Nonuniformly elliptic equations of \(p\)-Laplacian type, Nonlinear Anal., 61, 8, 1483-1495 (2005) · Zbl 1125.35344 |
| [26] | Flucher, M., Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 67, 3, 471-497 (1992) · Zbl 0763.58008 |
| [27] | Folland, G. B.; Stein, E. M., Estimates for the \(\overline{\partial}_b\) complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27, 429-522 (1974) · Zbl 0293.35012 |
| [28] | Folland, G. B.; Stein, E. M., (Hardy Spaces on Homogeneous Groups. Hardy Spaces on Homogeneous Groups, Math. Notes, vol. 28 (1982), Princeton University Press, University of Tokyo Press: Princeton University Press, University of Tokyo Press Princeton, NJ, Tokyo), xii+285 · Zbl 0508.42025 |
| [29] | Fontana, L., Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68, 415-454 (1993) · Zbl 0844.58082 |
| [30] | Fontana, L.; Morpurgo, C., Adams inequalities on measure spaces, Adv. Math., 226, 6, 5066-5119 (2011) · Zbl 1219.46032 |
| [31] | Franchi, B.; Gallot, S.; Wheeden, R. L., Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann., 300, 4, 557-571 (1994) · Zbl 0830.46027 |
| [32] | R.L. Frank, E.H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. of Math. (in press). arXiv:1009.1410v1; R.L. Frank, E.H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. of Math. (in press). arXiv:1009.1410v1 · Zbl 1252.42023 |
| [33] | X. Han, Existence of extremals of the Hardy-Littleowood-Sobolev inequality on the Heisenberg group, Indiana Univ. Math. J. (in press).; X. Han, Existence of extremals of the Hardy-Littleowood-Sobolev inequality on the Heisenberg group, Indiana Univ. Math. J. (in press). |
| [34] | Han, X.; Lu, G.; Zhu, J., Hardy-Littlewood-Sobolev and Stein-Weiss inequalities and integral systems on the Heisenberg group, Nonlinear Anal., 75, 11, 4296-4314 (2012) · Zbl 1309.42032 |
| [35] | Hardy, G. H.; Littlewood, G. E.; Polya, G., Inequalities (1934), Cambride Univ. Press: Cambride Univ. Press London · Zbl 0010.10703 |
| [36] | Jerison, D.; Lee, J. M., The Yamabe problem on CR manifolds, J. Differential Geom., 25, 167-197 (1987) · Zbl 0661.32026 |
| [37] | Jerison, D.; Lee, J. M., Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc., 1, 1-13 (1988) · Zbl 0634.32016 |
| [38] | Jerison, D.; Lee, J. M., Intrinsic CR coordinates and the CR Yamabe problem, J. Differential Geom., 29, 303-343 (1989) · Zbl 0671.32016 |
| [39] | Judovič, V. I., Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138, 805-808 (1961), (in Russian) · Zbl 0144.14501 |
| [40] | Lam, N.; Lu, G., Existence and multiplicity of solutions to equations of \(N\)-Laplacian type with critical exponential growth in \(R^N\), J. Funct. Anal., 262, 3, 1132-1165 (2012) · Zbl 1236.35050 |
| [41] | Lam, N.; Lu, G., Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dynam. Systems, 32, 6, 2187-2205 (2012) · Zbl 1245.35064 |
| [42] | Lam, N.; Lu, G., Sharp Adams type inequalities in Sobolev spaces \(W^{m, \frac{n}{m}}(R^n)\) for arbitrary interger \(m\), J. Differential Equations, 253, 4, 1143-1171 (2012) · Zbl 1259.46029 |
| [43] | N. Lam, G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument. Preprint.; N. Lam, G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument. Preprint. · Zbl 1294.46034 |
| [44] | Lam, N.; Lu, G., Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti Rabinowitz condition, J. Geom. Anal. (2012) |
| [45] | N. Lam, G. Lu, \(N \mathbb{R}^N\); N. Lam, G. Lu, \(N \mathbb{R}^N\) · Zbl 1283.35049 |
| [46] | Lam, N.; Lu, G., The Moser-Trudinger and Adams inequalities and elliptic and subelliptic equations with nonlinearity of exponential growth, (Recent Development in Geometry and Analysis. Recent Development in Geometry and Analysis, Advanced Lectures in Mathematics, vol. 23 (2012)), 179-251 · Zbl 1317.35018 |
| [47] | N. Lam, G. Lu, The sharp singular Adams inequalities in high order Sobolev spaces. arXiv:1112.6431v1; N. Lam, G. Lu, The sharp singular Adams inequalities in high order Sobolev spaces. arXiv:1112.6431v1 · Zbl 1319.46027 |
| [48] | Lam, N.; Lu, G.; Tang, H., On nonuniformly subelliptic equations of \(Q\)-sub-Laplacian type with critical growth in \(H^n\), Adv. Nonlinear Stud., 12, 659-681 (2012) · Zbl 1282.35139 |
| [49] | N. Lam, G. Lu, H. Tang, On sharp subcritical Moser-Trudinger inequality on the entire Heisenberg group and subelliptic PDEs. Preprint.; N. Lam, G. Lu, H. Tang, On sharp subcritical Moser-Trudinger inequality on the entire Heisenberg group and subelliptic PDEs. Preprint. · Zbl 1282.35393 |
| [50] | Li, Y. X., Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14, 2, 163-192 (2001) · Zbl 0995.58021 |
| [51] | Li, Y. X., Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48, 5, 618-648 (2005) · Zbl 1100.53036 |
| [52] | Li, Y. X.; Ndiaye, C., Extremal functions for Moser-Trudinger type inequality on compact closed 4-manifolds, J. Geom. Anal., 17, 4, 669-699 (2007) · Zbl 1132.58011 |
| [53] | Li, Y. X.; Ruf, B., A sharp Trudinger-Moser type inequality for unbounded domains in \(R^n\), Indiana Univ. Math. J., 57, 1, 451-480 (2008) · Zbl 1157.35032 |
| [54] | Lieb, E. H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118, 2, 349-374 (1983) · Zbl 0527.42011 |
| [55] | Lin, K. C., Extremal functions for Moser’s inequality, Trans. Amer. Math. Soc., 348, 7, 2663-2671 (1996) · Zbl 0861.49001 |
| [56] | Lions, P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I., Rev. Mat. Iberoam., 1, 1, 145-201 (1985) · Zbl 0704.49005 |
| [57] | Lions, P.-L., The concentration-compactness principle in the calculus of variations. The limit case. II., Rev. Mat. Iberoam., 1, 2, 45-121 (1985) · Zbl 0704.49006 |
| [58] | Lu, G., Weighted Poincare and Sobolev inequalities for vector fields satisfying Hormander’s condition and applications, Rev. Mat. Iberoam., 8, 3, 367-439 (1992) · Zbl 0804.35015 |
| [59] | Lu, G.; Wheeden, R. L., High order representation formulas and embedding theorems on stratified groups and generalizations, Studia Math., 142, 101-133 (2000) · Zbl 0974.46039 |
| [60] | Lu, G.; Yang, Y., Adams’ inequalities for bi-Laplacian and extremal functions in dimension four, Adv. Math., 220, 4, 1135-1170 (2009) · Zbl 1172.46021 |
| [61] | Lu, G.; Yang, Y., A sharpened Moser-Pohozaev-Trudinger inequality with mean value zero in \(R^2\), Nonlinear Anal., 70, 8, 2992-3001 (2009) · Zbl 1179.58010 |
| [62] | J.J. Manfredi, V.N. Vera De Serio, Rearrangements in Carnot groups. Preprint.; J.J. Manfredi, V.N. Vera De Serio, Rearrangements in Carnot groups. Preprint. · Zbl 1421.30034 |
| [63] | Moser, J., A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20, 1077-1092 (1970-1971) · Zbl 0203.43701 |
| [64] | Pohožaev, S. I., On the eigenfunctions of the equation \(\Delta u + \lambda f(u) = 0\), Dokl. Akad. Nauk SSSR, 165, 36-39 (1965), (in Russian) · Zbl 0141.30202 |
| [65] | Ruf, B., A sharp Trudinger-Moser type inequality for unbounded domains in \(R^2\), J. Funct. Anal., 219, 2, 340-367 (2005) · Zbl 1119.46033 |
| [66] | B. Ruf, F. Sani, Sharp Adams-type inequalities in \(\mathbb{R}^n\); B. Ruf, F. Sani, Sharp Adams-type inequalities in \(\mathbb{R}^n\) · Zbl 1280.46024 |
| [67] | Shaw, M. C., Eigenfunctions of the nonlinear equation \(\Delta u + \nu f(x, u) = 0\) in \(R^2\), Pacific J. Math., 129, 2, 349-356 (1987) · Zbl 0657.35107 |
| [68] | Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110, 353-372 (1976) · Zbl 0353.46018 |
| [69] | C. Tarsi, Adams’ Inequality and Limiting Sobolev Embeddings into Zygmund Spaces, Potential Anal. http://dx.doi.org/10.1007/s11118-011-9259-4; C. Tarsi, Adams’ Inequality and Limiting Sobolev Embeddings into Zygmund Spaces, Potential Anal. http://dx.doi.org/10.1007/s11118-011-9259-4 |
| [70] | Trudinger, N. S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17, 473-483 (1967) · Zbl 0163.36402 |
| [71] | J. Zhu, The improved Moser-Trudinger inequality with \(L^pn\); J. Zhu, The improved Moser-Trudinger inequality with \(L^pn\) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
