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Sharp weighted Trudinger–Moser–Adams inequalities on the whole space and the existence of their extremals

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Abstract

Though there have been extensive works on the existence of maximizers for sharp first order Trudinger–Moser inequalities, much less is known for that of the maximizers for higher order Adams’ inequalities. In this paper, we mainly study the existence of extremals for sharp weighted Trudinger–Moser–Adams type inequalities with the Dirichlet and Sobolev norms (also known as the critical and subcritical Trudinger–Moser–Adams inequalities), see Theorems 1.1, 1.2, 1.3, 1.5, 1.7, 1.9 and 1.11. First, we employ the method based on level-sets of functions under consideration and Fourier transform to establish stronger weighted Trudinger–Moser–Adams type inequalities with the Dirichlet norm in \(W^{2,\frac{n}{2}}(\mathbb {R}^n)\) and \(W^{m,2}(\mathbb {R}^{2m})\) respectively. While the first order sharp weighted Trudinger–Moser inequality and its existence of extremal functions was established by Dong and the second author using a quasi-conformal type transform (Dong and Lu in Calc Var Partial Differ Equ 55:55–88, 2016), such a transform does not work for the Adams inequality involving higher order derivatives. Since the absence of the Polyá–Szegö inequality and the failure of change of variable method for higher order derivatives for weighted inequalities, we will need several compact embedding results (Lemmas 2.1, 3.1 and 5.2). Through the compact embedding and scaling invariance of the subcritical Adams inequality, we investigate the attainability of best constants. Second, we employ the method developed by Lam et al. (Adv Math 352:1253–1298, 2019) which uses the relationship between the supremums of the critical and subcritical inequalities (see also Lam in Proc Amer Math Soc 145:4885–4892, 2017) to establish the existence of extremals for weighted Adams’ inequalities with the Sobolev norm. Third, using the Fourier rearrangement inequality established by Lenzmann and Sok (A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, arXiv:1805.06294v1), we can reduce our problem to the radial case and then establish the existence of the extremal functions for the non-weighted Adams inequalities. As an application, we derive new results on high-order critical Caffarelli–Kohn–Nirenberg interpolation inequalities whose parameters extend those proved by Lin (Commun Partial Differ Equ 11:1515–1538, 1986) (see Theorems 1.13 and 1.14). Furthermore, we also establish the relationship between the best constants of the weighted Trudinger–Moser–Adams type inequalities and the Caffarelli–Kohn–Nirenberg inequalities in the asymptotic sense (see Theorems 1.13 and 1.14).

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Notes

  1. We note that the main results of [33] were described in [13].

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  Google Scholar 

  2. Adams, D.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128, 383–398 (1998)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Cao, D.: Nontrivial solution of semilinear elliptic equation with critical exponent in \(\mathbb{R}^2\). Commun. Partial Differ. Equ. 17, 407–435 (1992)

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Chen, L., Li, J., Lu, G., Zhang, C.: Sharpened Adams inequality and ground state solutions to the bi-Laplacian equation in \(\mathbb{R}^4\). Adv. Nonlinear Stud. 18(3), 429–452 (2018)

    Article  MathSciNet  Google Scholar 

  8. Csató, G., Roy, P.: Extremal functions for the singular Moser–Trudinger inequality in 2 dimensions. Calc. Var. Partial Differ. Equ. 54, 2341–2366 (2015)

    Article  MathSciNet  Google Scholar 

  9. Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalies with weights. Compos. Math. 53, 259–275 (1984)

    MATH  Google Scholar 

  10. do Ó, J.M.: N-Laplacian equations in \(\mathbb{R}^{n}\) with critical growth. Abstr. Appl. Anal. 2, 301–315 (1997)

    Article  MathSciNet  Google Scholar 

  11. Dong, M.: Existence of extremal functions for higher-order Caffarelli–Kohn–Nirenberg inequalities. Adv. Nonlinear Stud. 18(3), 543–553 (2018)

    Article  MathSciNet  Google Scholar 

  12. Dong, M., Lam, N., Lu, G.: Sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities and their extremal functions. Nonlinear Anal. 173, 75–98 (2018)

    Article  MathSciNet  Google Scholar 

  13. Dong, M., Lu, G.: Best constants and existence of maximizers for weighted Trudinger-Moser inequalities. Calc. Var. Partial Differ. Equ. 55(4), 26 (2016)

    Article  MathSciNet  Google Scholar 

  14. Flucher, M.: Extremal functions for the Trudinger–Moser inequality in 2 dimensions. Comment. Math. Helv. 67, 471–497 (1992)

    Article  MathSciNet  Google Scholar 

  15. Fontana, L., Morpurgo, C.: Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on \(\mathbb{R}^n\). Nonlinear Anal. 167, 85–122 (2018)

    Article  MathSciNet  Google Scholar 

  16. Ibrahim, S., Masmoudi, N., Nakanishi, K.: Trudinger–Moser inequality on the whole plane with the exact growth condition. J. Eur. Math. Soc. (JEMS) 17, 819–835 (2015)

    Article  MathSciNet  Google Scholar 

  17. Ishiwata, M.: Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in \(\mathbb{R}^N\). Math. Ann. 351, 781–804 (2011)

    Article  MathSciNet  Google Scholar 

  18. Ishiwata, M., Nakamura, M., Wadade, H.: On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form. Ann. Inst. Henri Poincar Anal. Non Linaire 31, 297–314 (2014)

    Article  MathSciNet  Google Scholar 

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

  20. 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, 1951–1974 (2006)

    Article  MathSciNet  Google Scholar 

  21. Lam, N.: Maximizers for the singular Trudinger–Moser inequalities in the subcritical cases. Proc. Amer. Math. Soc. 145, 4885–4892 (2017)

    Article  MathSciNet  Google Scholar 

  22. Lam, N.: Equivalence of sharp Trudinger–Moser–Adams inequalities. Commun. Pure Appl. Anal. 16(3), 973–997 (2017)

    Article  MathSciNet  Google Scholar 

  23. Lam, N.: Sharp subcritical and critical Trudinger–Moser inequalities on \(R^2\) and their extremal functions. Potential Anal. 46(1), 75–103 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Lam, N., Lu, G., Tang, H.: Sharp affine and improved Moser–Trudinger–Adams type inequalities on unbounded domains in the spirit of Lions. J. Geom. Anal. 27(1), 300–334 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  28. Lam, N., Lu, G.: Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications. Adv. Math. 231(6), 3259–3287 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Lam, N., Lu, G.: Sharp constants and optimizers for a class of the Caffarelli–Kohn–Nirenberg inequalities. Adv. Nonlinear Stud. 17, 457–480 (2017)

    Article  MathSciNet  Google Scholar 

  31. Lam, N., Lu, G.: 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, 13, pp. 43–80. Springer, Cham (2015)

  32. Lam, N., Lu, G., Zhang, L.: Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities. Rev. Mat. Iberoam. 33, 1219–1246 (2017)

    Article  MathSciNet  Google Scholar 

  33. Lam, N., Lu, G., Zhang, L.: Existence and nonexistence of extremal functions for sharp Trudinger-Moser-inequalities. Adv. Math. 352, 1253–1298 (2019)

    Article  MathSciNet  Google Scholar 

  34. Lam, N., Lu, G., Zhang, L.: Sharp singular Trudinger–Moser inequalities under different norms. Adv. Nonlinear Stud. 19(2), 239–261 (2019)

    Article  MathSciNet  Google Scholar 

  35. Lenzmann, E., Sok, J.: A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order. arXiv:1805.06294v1

  36. Li, J., Lu, G., Yang, Q.: Fourier analysis and optimal Hardy–Adams inequalities on hyperbolic spaces of any even dimension. Adv. Math. 333, 350–385 (2018)

    Article  MathSciNet  Google Scholar 

  37. Li, J., Lu, G., Zhu, M.: Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions. Calc. Var. Partial Differ. Equ. 57(3), Art. 84 (2018)

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. Lin, C.S.: Interpolation inequalities with weights. Commun. Partial Differ. Equ. 11, 1515–1538 (1986)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  42. Lu, G., Tang, H.: Sharp Moser–Trudinger inequalities on hyperbolic spaces with exact growth condition. J. Geom. Anal. 26(2), 837–857 (2016)

    Article  MathSciNet  Google Scholar 

  43. Lu, G., Tang, H.: Sharp singular Trudinger–Moser inequalities in Lorentz–Sobolev spaces. Adv. Nonlinear Stud. 16(3), 581–601 (2016)

    Article  MathSciNet  Google Scholar 

  44. Lu, G., Tang, H., Zhu, M.: Best constants for Adams’ inequalities with the exact growth condition in \(R^n\). Adv. Nonlinear Stud. 15(4), 763–788 (2015)

    Article  MathSciNet  Google Scholar 

  45. Lu, G., Yang, Q.: Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four. Adv. Math. 319, 567–598 (2017)

    Article  MathSciNet  Google Scholar 

  46. Lu, G., Yang, Q.: A sharp Trudinger–Moser inequality on any bounded and convex planar domain. Calc. Var. Partial Differ. Equ. 55(6), Art. 153 (2016)

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

    Article  MathSciNet  Google Scholar 

  48. Lu, G., Zhu, M.: A sharp Trudinger–Moser type inequality involving \(L^n\) norm in the entire space \(R^n\). J. Differ. Equ. 267(5), 3046–3082 (2019)

    Article  Google Scholar 

  49. Malchiodi, A., Martinazzi, L.: Critical points of the Moser–Trudinger functional on a disk. J. Eur. Math. Soc. 16, 893–908 (2014)

    Article  MathSciNet  Google Scholar 

  50. Malchiodi, A., Ruiz, D.: New improved Moser–Trudinger inequalities and singular Liouville equations on compact surfaces. Geom. Funct. Anal. 21, 1196–1217 (2011)

    Article  MathSciNet  Google Scholar 

  51. Masmoudi, N., Sani, F.: Adams’ inequality with the exact growth condition in \(\mathbb{R}^4\). Commun. Pure Appl. Math. 67, 1307–1335 (2014)

    Article  Google Scholar 

  52. Masmoudi, N., Sani, F.: Trudinger–Moser inequalities with the exact growth condition in \(\mathbb{R}^N\) and applications. Commun. Partial Differ. Equ. 40, 1408–1440 (2015)

    Article  Google Scholar 

  53. Masmoudi, N., Sani, F.: Higher order Adams’ inequality with the exact growth condition. Commun. Contemp. Math. 20, 33 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  55. Pohozaev, S.I.: The Sobolev embedding in the special case \(pl=n\). In: Proceeding of the Technical Scientific Conference on Advances of Scientific Research 1964–1965. Mathematics Sections Moscov. Eberget. Inst. Moscow, pp. 158–170 (1965)

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  58. Tang, H.: Equivalence of sharp Trudinger–Moser inequalities in Lorentz–Sobolev spaces. Potential Anal. (2019). https://doi.org/10.1007/s11118-019-09769-9

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  61. Wang, X.: Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Commun. Pure Appl. Anal. 18(5), 2717–2733 (2019)

    Article  MathSciNet  Google Scholar 

  62. Wang, X.: Improved Hardy–Adams inequality on hyperbolic space of dimension four. Nonlinear Anal. 182, 45–56 (2019)

    Article  MathSciNet  Google Scholar 

  63. Zhang, C., Chen, L.: Concentration-compactness principle of singular Trudinger–Moser inequalities in \(R^n\) and \(n\)-Laplace equations. Adv. Nonlinear Stud. 18(3), 567–585 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank the referee for her/his many comments which help to improve the exposition of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Lu Chen

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Guozhen Lu

  3. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, People’s Republic of China

    Caifeng Zhang

  4. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Caifeng Zhang

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  1. Lu Chen
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Correspondence to Guozhen Lu or Caifeng Zhang.

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Guozhen Lu was partly supported by a Simons Collaboration Grant from the Simons Foundation.

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Chen, L., Lu, G. & Zhang, C. Sharp weighted Trudinger–Moser–Adams inequalities on the whole space and the existence of their extremals. Calc. Var. 58, 132 (2019). https://doi.org/10.1007/s00526-019-1580-6

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