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Sharp Singular Trudinger-Moser-Adams Type Inequalities with Exact Growth

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Geometric Methods in PDE’s

Part of the book series: Springer INdAM Series ((SINDAMS,volume 13))

Abstract

The main purpose of this paper is two fold. On the one hand, we review some recent progress on best constants for various sharp Moser-Trudinger and Adams inequalities in Euclidean spaces \(\mathbb{R}^{N}\), hyperbolic spaces and other settings, and such sharp inequalities of Lions type. On the other hand, we present and prove some new results on sharp singular Moser-Trudinger and Adams type inequalities with exact growth condition and their affine analogues of such inequalities (Theorems 1.1, 1.2 and 1.3). We also establish a sharpened version of the classical Moser-Trudinger inequality on finite balls (Theorem 1.4).

Dedicated to Ermanno Lanconelli on the occasion of his 70th birthday, with friendship

Mathematics Subject Classification: 26D10, 46E35

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References

  1. Adachi, S., Tanaka, K.: Trudinger type inequalities in \(\mathbb{R}^{N}\) and their best exponents. Proc. Am. Math. Soc. 128, 2051–2057 (1999)

    Article  MathSciNet  Google Scholar 

  2. Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. (2) 128(2), 385–398 (1988)

    Google Scholar 

  3. Adimurthi, A., Druet, O.: Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality. Commun. Partial Differ. Equ. 29(1–2), 295–322 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Adimurthi, A., Sandeep, K.: A singular Moser-Trudinger embedding and its applications. NoDEA Nonlinear Differ. Equ. Appl. 13(5–6), 585–603 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Adimurthi, A., Yang, Y.: An interpolation of Hardy inequality and Trudinger-Moser inequality in \(\mathbb{R}^{N}\) and its applications. Int. Math. Res. Not. IMRN 13, 2394–2426 (2010)

    MathSciNet  Google Scholar 

  6. Balogh, Z., 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Beckner, W.: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. Math. (2) 138(1), 213–242 (1993)

    Google Scholar 

  8. Branson, T., Fontana, L., Morpurgo, C.: Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere. Ann. Math. (2) 177(1), 1–52 (2013)

    Google Scholar 

  9. Cao, D.: Nontrivial solution of semilinear elliptic equation with critical exponent in R 2. Commun. Partial Differ. Equ. 17(3–4), 407–435 (1992)

    Article  MATH  Google Scholar 

  10. Carleson, L., Chang, S.Y.A.: On the existence of an extremal function for an inequality. J. Moser. Bull. Sci. Math. (2) 110(2), 113–127 (1986)

    Google Scholar 

  11. Cassani, D., Tarsi, C.: A Moser-type inequality in Lorentz-Sobolev spaces for unbounded domains in \(\mathbb{R}^{N}\). Asymptot. Anal. 64(1–2), 29–51 (2009)

    MATH  MathSciNet  Google Scholar 

  12. Chang, S.Y.A., Yang, P.: Conformal deformation of metrics on S 2. J. Differ. Geom. 27, 259–296 (1988)

    MATH  MathSciNet  Google Scholar 

  13. Chang, S.Y.A., Yang, P.: The inequality of Moser and Trudinger and applications to conformal geometry. Dedicated to the memory of Jürgen K. Moser. Commun. Pure Appl. Math. 56(8), 1135–1150 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cianchi, A.: Moser-Trudinger inequalities without boundary conditions and isoperimetric problems. Indiana Univ. Math. J. 54(3), 669–705 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cianchi, A.: Moser-Trudinger trace inequalities. Adv. Math. 217(5), 2005–2044 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cianchi, A., Lutwak, E., Yang, D., Zhang, G.: Affine Moser-Trudinger and Morrey-Sobolev inequalities. Calc. Var. 36(3), 419–436 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cohn, W.S., Lu, G.: Best constants for Moser-Trudinger inequalities on the Heisenberg group. Indiana Univ. Math. J. 50(4), 1567–1591 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. 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)

    Google Scholar 

  19. Cohn, W.S., Lu, G.: Sharp constants for Moser-Trudinger inequalities on spheres in complex space C n. Commun. Pure Appl. Math. 57(11), 1458–1493 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. 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. Theory Methods Appl. 75(12), 4483–4495 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. de Figueiredo, D.G., do Ó, J.M., Ruf, B.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Commun. Pure Appl. Math. 55(2), 135–152 (2002)

    Google Scholar 

  22. do Ó, J.M.: N-Laplacian equations in R N with critical growth. Abstr. Appl. Anal. 2(3–4), 301–315 (1997)

    Google Scholar 

  23. Flucher, M.: Extremal functions for the Trudinger-Moser inequality in 2 dimensions. Comment. Math. Helv. 67 (1992), no. 3, 471–497.

    Article  MATH  MathSciNet  Google Scholar 

  24. Fontana, L.: Sharp borderline Sobolev inequalities on compact Riemannian manifolds. Commun. Math. Helv. 68, 415–454 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  25. Fontana, L., Morpurgo, C.: Adams inequalities on measure spaces. Adv. Math. 226(6), 5066–5119 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Haberl, C., Schuster, F.E., Xiao, J.: An asymmetric affine Pólya–Szegö principle. Math. Ann. 352(3), 517–542

    Google Scholar 

  27. Ibrahim, S., Masmoudi, N., Nakanishi, K.: Trudinger-Moser inequality on the whole plane with the exact growth condition. arXiv:1110.1712v1

    Google Scholar 

  28. Ishiwata, M.: Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in R N. Math. Ann. 351(4), 781–804 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ishiwata, M., Nakamura, M., Wadade, H.: On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form. Ann. I. H. Poincaré-AN (2013). http://dx.doi.org/10.1016/j.anihpc.2013.03.004

  30. Kozono, H., Sato, T., Wadade, H.: Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo-Nirenberg inequality. Indiana Univ. Math. J. 55(6), 1951–1974 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lam, N., Lu, G.: The Moser-Trudinger and Adams inequalities and elliptic and subelliptic equations with nonlinearity of exponential growth. In: Recent Developments in Geometry and Analysis. Advanced Lectures in Mathematics (ALM), vol. 23, pp. 179–251. International Press, Somerville (2012)

    Google Scholar 

  32. Lam, N., Lu, G.: Sharp Adams type inequalities in Sobolev spaces \(W^{m, \frac{n} {m} }\left (\mathbb{R}^{n}\right )\) for arbitrary integer m. J. Differ. Equ. 253, 1143–1171 (2012)

    Google Scholar 

  33. Lam, N., Lu, G.: Sharp singular Adams inequalities in high order Sobolev spaces. Methods Appl. Anal. 19(3), 243–266 (2012)

    MATH  MathSciNet  Google Scholar 

  34. Lam, N., Lu, G.: Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications. Adv. Math. 231, 3259–3287 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lam, N., Lu, G.: A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument. J. Differ. Equ. 255(3), 298–325 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  36. Lam, N., Lu, G., Zhang, L.: Sharp singular Trudinger-Moser inequalities and their extremals. Preprint (2013)

    Google Scholar 

  37. Lam, N., Tang, H.: Sharp constants for weighted Moser-Trudinger inequalities on groups of Heisenberg type. Nonlinear Anal. 89, 95–109 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  38. Lam, N., Lu, G., Tang, H.: On nonuniformly subelliptic equations of Q-sub-Laplacian type with critical growth in the Heisenberg group. Adv. Nonlinear Stud. 12(3), 659–681 (2012)

    MATH  MathSciNet  Google Scholar 

  39. Lam, N., Lu, G., Tang, H.: Sharp affine and improved Moser-Trudinger-Adams type inequalities on unbounded domains in the spirit of Lions. Preprint (2013)

    Google Scholar 

  40. Lam, N., Lu, G., Tang, H.: Sharp subcritical Moser-Trudinger inequalities on Heisenberg groups and subelliptic PDEs. Nonlinear Anal. 95, 77–92 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  41. Leckband, M.: Moser’s inequality on the ball B n for functions with mean value zero. Commun. Pure Appl. Math. LVIII, 0789–0798 (2005)

    Google Scholar 

  42. Li, Y.X.: Moser-Trudinger inequality on compact Riemannian manifolds of dimension two. J. Partial Differ. Equ. 14(2), 163–192 (2001)

    MATH  Google Scholar 

  43. Li, Y.X.: Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds. Sci. China Ser. A 48(5), 618–648 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  44. 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)

    Article  MATH  MathSciNet  Google Scholar 

  45. Li, Y.X., Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in \(\mathbb{R}^{n}\). Indiana Univ. Math. J. 57(1), 451–480 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  46. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, xxii+346 pp, 2nd edn. American Mathematical Society, Providence (2001)

    Google Scholar 

  47. Lin, K.: Extremal functions for Moser’s inequality. Trans. Am. Math. Soc. 348(7), 2663–2671 (1996)

    Article  MATH  Google Scholar 

  48. Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoam. 1(2), 45–121 (1985)

    Article  MATH  Google Scholar 

  49. Lu, G., Tang, H.: Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces. Adv. Nonlinear Stud. 13, 1035–1052 (2013)

    MATH  MathSciNet  Google Scholar 

  50. Lu, G., Tang, H.: Sharp Moser-Trudinger inequalities on hyperbolic spaces with the exact growth condition. J. Geom. Anal. (2015)

    Google Scholar 

  51. Lu, G., Yang, Y.: A sharpened Moser-Pohozaev-Trudinger inequality with mean value zero in R 2. Nonlinear Anal. 70(8), 2992–3001 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  52. Lu, G., Yang, Y.: Adams’ inequalities for bi-Laplacian and extremal functions in dimension four. Adv. Math. 220, 1135–1170 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  53. Lu, G., Tang, H., Zhu, M.: Sharp Trudinger-Moser-Adams inequality with the exact growth condition in R n. Adv. Nonlinear Stud. 15(4) (2015)

    Google Scholar 

  54. Masmoudi, N., Sani, F.: Adams’ inequality with the exact growth condition in R 4. Commun. Pure Appl. Math. 67(8), 1307–1335 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  55. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/1971)

    Google Scholar 

  56. Ogawa, T.: A proof of Trudinger’s inequality and its application to nonlinear Schršdinger equations. Nonlinear Anal. 14(9), 765–769 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  57. Ozawa, T.: On critical cases of Sobolev’s inequalities. J. Funct. Anal. 127(2), 259–269 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  58. Pohožaev, S.I.: On the eigenfunctions of the equation \(\varDelta u +\lambda f(u) = 0\). (Russian) Dokl. Akad. Nauk SSSR 165, 36–39 (1965)

    Google Scholar 

  59. Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in \(\mathbb{R}^{2}\). J. Funct. Anal. 219(2), 340–367 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  60. Ruf, B., Sani, F.: Sharp Adams-type inequalities in \(\mathbb{R}^{n}\). Trans. Am. Math. Soc. 365(2), 645–670 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  61. Tarsi, C.: Adams’ inequality and limiting Sobolev embeddings into Zygmund spaces. Potential Anal. 37(4), 353–385 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  62. Tian, G., Wang, X.: Moser-Trudinger type inequalities for the Hessian equation. J. Funct. Anal. 259(8), 1974–2002 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  63. Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)

    MATH  MathSciNet  Google Scholar 

  64. Wang, G., Ye, D.: A Hardy-Moser-Trudinger inequality. Adv. Math. 230(1), 294–320 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  65. Yudovič, V. I.: Some estimates connected with integral operators and with solutions of elliptic equations. (Russian) Dokl. Akad. Nauk SSSR 138, 805–808 (1961)

    Google Scholar 

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Acknowledgements

This research is partly supported by a US NSF grant DMS#1301595.

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, USA

    Nguyen Lam & Guozhen Lu

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  1. Nguyen Lam
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  2. Guozhen Lu
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Correspondence to Guozhen Lu .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Bologna, Bologna, Italy

    Giovanna Citti

  2. Dipartimento di Matematica, Università di Bologna, Bologna, Bologna, Italy

    Maria Manfredini

  3. Dipartimento di Matematica, Università di Bologna, Bologna, Bologna, Italy

    Daniele Morbidelli

  4. Dipartimento FIM, Università di Modena e Reggio Emilia, Modena, Italy

    Sergio Polidoro

  5. Dipartimento di Matematica, Università di Bologna, Bologna, Italy

    Francesco Uguzzoni

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Lam, N., Lu, G. (2015). Sharp Singular Trudinger-Moser-Adams Type Inequalities with Exact Growth. In: Citti, G., Manfredini, M., Morbidelli, D., Polidoro, S., Uguzzoni, F. (eds) Geometric Methods in PDE’s. Springer INdAM Series, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-02666-4_3

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