July  2020, 19(7): 3559-3574. doi: 10.3934/cpaa.2020155

The Hardy–Moser–Trudinger inequality via the transplantation of Green functions

Department of Mathematics, FPT University, Ha Noi, Vietnam

Received  April 2019 Revised  January 2020 Published  April 2020

We provide a new proof of the Hardy–Moser–Trudinger inequality and the existence of its extremals which are established by Wang and Ye ("G. Wang, and D. Ye, A Hardy–Moser–Trudinger inequality, Adv. Math, 230 (2012) 294–230.") without using the blow-up analysis method. Our proof is based on the transformation of functions via the transplantation of Green functions. This method enables us to compute explicitly the concentrating level of the Hardy–Moser–Trudinger functional over the normalizing concentrating sequences which is crucial to prove the existence of extremals for the Hardy–Moser–Trudinger inequality. Some comments on the applications of this approach to some other Moser–Trudinger type inequalities are given.

Citation: Van Hoang Nguyen. The Hardy–Moser–Trudinger inequality via the transplantation of Green functions. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3559-3574. doi: 10.3934/cpaa.2020155
References:
[1]

S. Adachi and K. Tanaka, Trudinger type inequalities in ${\mathbb R}^N$ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.  doi: 10.1090/S0002-9939-99-05180-1.  Google Scholar

[2]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. Math., 128 (1988), 385-398.  doi: 10.2307/1971445.  Google Scholar

[3]

Adimurthi and O. Druet, Blow-up analysis in dimension $2$ and a sharp form of Trudinger-Moser inequality, Commun. Partial Differ. Equ., 29 (2004), 295-322.  doi: 10.1081/PDE-120028854.  Google Scholar

[4]

Ad imurthi and K. Sandeep, A singular Moser–Trudinger embedding and its applications, NoDea Nonlinear Differ. Equ. Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[5]

Adimurthi and C. Tintarev, On a version of Trudinger-Moser inequality with Möbius shift invariance, Calc. Var. Partial Differ. Equ., 39 (2010), 203-212.  doi: 10.1007/s00526-010-0307-5.  Google Scholar

[6]

Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger–Moser inequality in ${\mathbb R}^N$ and its applications, Int. Math. Res. Not. IMRN, (2010), 2394–2426. doi: 10.1093/imrn/rnp194.  Google Scholar

[7]

R. D. BenguriaR. L. Frank and M. Loss, The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space, Math. Res. Lett., 15 (2008), 613-622.  doi: 10.4310/MRL.2008.v15.n4.a1.  Google Scholar

[8]

L. Carleson and S. Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127.   Google Scholar

[9]

G. Csató and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in $2$ dimensions, Calc. Var. Partial Differ. Equ., 54 (2015), 2341-2366.  doi: 10.1007/s00526-015-0867-5.  Google Scholar

[10]

G. Csató and P. Roy, Singular Moser-Trudinger inequality on simply connected domains, Commun. Partial Differ. Equ., 41 (2016), 838-847.  doi: 10.1080/03605302.2015.1123276.  Google Scholar

[11]

G. Csató, V. H. Nguyen and P. Roy, Extremals for the singular Moser-Trudinger inequality via $n$-harmonic transplantation, preprint, arXiv: 1801.03932v3. Google Scholar

[12]

D. G. De FigueiredoJ. M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Commun. Pure Appl. Math., 55 (2002), 135-152.  doi: 10.1002/cpa.10015.  Google Scholar

[13]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv., 67 (1992), 471-497.  doi: 10.1007/BF02566514.  Google Scholar

[14]

M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser inequalities in $\mathbb R^N$, Math. Ann., 351 (2011), 781-804.  doi: 10.1007/s00208-010-0618-z.  Google Scholar

[15]

N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: A rearrangement–free argument, J. Differ. Equ., 255 (2013), 298-325.  doi: 10.1016/j.jde.2013.04.005.  Google Scholar

[16]

Y. Li, Moser-Trudinger inequaity on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.   Google Scholar

[17]

Y. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648.  doi: 10.1360/04ys0050.  Google Scholar

[18]

Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb R}^n$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[19]

J. LiG. Lu and Q. Yang, Fourier analysis and optimal Hardy-Adams inequalities on hyperbolic spaces of any even dimension, Adv. Math., 333 (2018), 350-385.  doi: 10.1016/j.aim.2018.05.035.  Google Scholar

[20]

K. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671.  doi: 10.1090/S0002-9947-96-01541-3.  Google Scholar

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoam., 1 (1985), 45-121.  doi: 10.4171/RMI/12.  Google Scholar

[22]

G. Lu and Q. Yang, A sharp Trudinger-Moser inequality on any bounded and convex planar domain, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 153, 16 pp. doi: 10.1007/s00526-016-1077-5.  Google Scholar

[23]

G. Lu and Q. Yang, Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four, Adv. Math., 319 (2017), 567-598.  doi: 10.1016/j.aim.2017.08.014.  Google Scholar

[24]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.  doi: 10.1142/S0219199710004111.  Google Scholar

[25]

G. ManciniK. Sandeep and C. Tintarev, Trudinger-Moser inequality in the hyperbolic space $\mathbb H^n$, Adv. Nonlinear Anal., 2 (2013), 309-324.  doi: 10.1515/anona-2013-0001.  Google Scholar

[26]

G. Mancini and L. Martinazzi, The Moser-Trudinger inequality and its extremals on a disk via energy estimates, Calc. Var. Partial Differ. Equ., 56 (2017), Art. 94, 26 pp. doi: 10.1007/s00526-017-1184-y.  Google Scholar

[27]

L. Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.  doi: 10.1016/j.na.2015.06.034.  Google Scholar

[28]

V. G. Maz'ya, Sobolev spaces, Springer Verlag, Berlin, New York, 1985. doi: 10.1007/978-3-662-09922-3.  Google Scholar

[29]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[30]

V. H. Nguyen, The sharp Poincaré–Sobolev type inequalities in the hyperbolic spaces $\mathbb H^n$, J. Math. Anal. Appl., 462 (2018), 1570-1584.  doi: 10.1016/j.jmaa.2018.02.054.  Google Scholar

[31]

V. H. Nguyen, Improved Moser-Trudinger type inequalities in the hyperbolic space $\mathbb H^n$, Nonlinear Anal., 168 (2018), 67-80.  doi: 10.1016/j.na.2017.11.009.  Google Scholar

[32]

V. H. Nguyen, Improved Moser–Trudinger inequality of Tintarev type in dimension $n$ and the existence of its extremal functions, Ann. Global Anal. Geom., 54 (2018), 237-256.  doi: 10.1007/s10455-018-9599-z.  Google Scholar

[33]

V. H. Nguyen, Improved singular Moser-Trudinger inequalities and their extremal functions, Potential Anal., in press. Google Scholar

[34]

V. H. Nguyen, The sharp Hardy–Moser–Trudinger inequality in dimension $n$, preprint, arXiv: 1909.12587. Google Scholar

[35]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$] (Russian), Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.   Google Scholar

[36]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb R}^2$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[37]

A. Tertikas and C. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality, Ann. Mat. Pura Appl. (4), 186 (2007), 645-662. doi: 10.1007/s10231-006-0024-z.  Google Scholar

[38]

C. Tintarev, Trudinger–Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66.  doi: 10.1016/j.jfa.2013.09.009.  Google Scholar

[39]

N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[40]

G. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Adv. Math., 230 (2012), 294-320.  doi: 10.1016/j.aim.2011.12.001.  Google Scholar

[41]

X. Wang, Improved Hardy-Adams inequality on hyperbolic space of dimension four, Nonlinear Anal., 182 (2019), 45-56.  doi: 10.1016/j.na.2018.12.007.  Google Scholar

[42]

X. Wang, Singular Hardy-Moser-Trudinger inequality and the existence of extremals on the unit disc, Commun. Pure Appl. Anal., 18 (2019), 2741-2757.  doi: 10.3934/cpaa.2019121.  Google Scholar

[43]

Y. Yang, A sharp form of the Moser–Trudinger inequality on a compact Riemannian surface, Trans. Amer. Math. Soc., 359 (2007), 5761-5776.  doi: 10.1090/S0002-9947-07-04272-9.  Google Scholar

[44]

Y. Yang and X. Zhu, An improved Hardy-Trudinger-Moser inequality, Ann. Global Anal. Geom., 49 (2016), 23-41.  doi: 10.1007/s10455-015-9478-9.  Google Scholar

[45]

Q. YangD. Su and Y. Kong, Sharp Moser-Trudinger inequalities on Riemannian manifolds with negative curvature, Ann. Mat. Pura Appl., 195 (2016), 459-471.  doi: 10.1007/s10231-015-0472-4.  Google Scholar

[46]

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

show all references

References:
[1]

S. Adachi and K. Tanaka, Trudinger type inequalities in ${\mathbb R}^N$ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.  doi: 10.1090/S0002-9939-99-05180-1.  Google Scholar

[2]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. Math., 128 (1988), 385-398.  doi: 10.2307/1971445.  Google Scholar

[3]

Adimurthi and O. Druet, Blow-up analysis in dimension $2$ and a sharp form of Trudinger-Moser inequality, Commun. Partial Differ. Equ., 29 (2004), 295-322.  doi: 10.1081/PDE-120028854.  Google Scholar

[4]

Ad imurthi and K. Sandeep, A singular Moser–Trudinger embedding and its applications, NoDea Nonlinear Differ. Equ. Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[5]

Adimurthi and C. Tintarev, On a version of Trudinger-Moser inequality with Möbius shift invariance, Calc. Var. Partial Differ. Equ., 39 (2010), 203-212.  doi: 10.1007/s00526-010-0307-5.  Google Scholar

[6]

Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger–Moser inequality in ${\mathbb R}^N$ and its applications, Int. Math. Res. Not. IMRN, (2010), 2394–2426. doi: 10.1093/imrn/rnp194.  Google Scholar

[7]

R. D. BenguriaR. L. Frank and M. Loss, The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space, Math. Res. Lett., 15 (2008), 613-622.  doi: 10.4310/MRL.2008.v15.n4.a1.  Google Scholar

[8]

L. Carleson and S. Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127.   Google Scholar

[9]

G. Csató and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in $2$ dimensions, Calc. Var. Partial Differ. Equ., 54 (2015), 2341-2366.  doi: 10.1007/s00526-015-0867-5.  Google Scholar

[10]

G. Csató and P. Roy, Singular Moser-Trudinger inequality on simply connected domains, Commun. Partial Differ. Equ., 41 (2016), 838-847.  doi: 10.1080/03605302.2015.1123276.  Google Scholar

[11]

G. Csató, V. H. Nguyen and P. Roy, Extremals for the singular Moser-Trudinger inequality via $n$-harmonic transplantation, preprint, arXiv: 1801.03932v3. Google Scholar

[12]

D. G. De FigueiredoJ. M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Commun. Pure Appl. Math., 55 (2002), 135-152.  doi: 10.1002/cpa.10015.  Google Scholar

[13]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv., 67 (1992), 471-497.  doi: 10.1007/BF02566514.  Google Scholar

[14]

M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser inequalities in $\mathbb R^N$, Math. Ann., 351 (2011), 781-804.  doi: 10.1007/s00208-010-0618-z.  Google Scholar

[15]

N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: A rearrangement–free argument, J. Differ. Equ., 255 (2013), 298-325.  doi: 10.1016/j.jde.2013.04.005.  Google Scholar

[16]

Y. Li, Moser-Trudinger inequaity on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.   Google Scholar

[17]

Y. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648.  doi: 10.1360/04ys0050.  Google Scholar

[18]

Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb R}^n$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[19]

J. LiG. Lu and Q. Yang, Fourier analysis and optimal Hardy-Adams inequalities on hyperbolic spaces of any even dimension, Adv. Math., 333 (2018), 350-385.  doi: 10.1016/j.aim.2018.05.035.  Google Scholar

[20]

K. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671.  doi: 10.1090/S0002-9947-96-01541-3.  Google Scholar

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoam., 1 (1985), 45-121.  doi: 10.4171/RMI/12.  Google Scholar

[22]

G. Lu and Q. Yang, A sharp Trudinger-Moser inequality on any bounded and convex planar domain, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 153, 16 pp. doi: 10.1007/s00526-016-1077-5.  Google Scholar

[23]

G. Lu and Q. Yang, Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four, Adv. Math., 319 (2017), 567-598.  doi: 10.1016/j.aim.2017.08.014.  Google Scholar

[24]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.  doi: 10.1142/S0219199710004111.  Google Scholar

[25]

G. ManciniK. Sandeep and C. Tintarev, Trudinger-Moser inequality in the hyperbolic space $\mathbb H^n$, Adv. Nonlinear Anal., 2 (2013), 309-324.  doi: 10.1515/anona-2013-0001.  Google Scholar

[26]

G. Mancini and L. Martinazzi, The Moser-Trudinger inequality and its extremals on a disk via energy estimates, Calc. Var. Partial Differ. Equ., 56 (2017), Art. 94, 26 pp. doi: 10.1007/s00526-017-1184-y.  Google Scholar

[27]

L. Martinazzi, Fractional Adams-Moser-Trudinger type inequalities, Nonlinear Anal., 127 (2015), 263-278.  doi: 10.1016/j.na.2015.06.034.  Google Scholar

[28]

V. G. Maz'ya, Sobolev spaces, Springer Verlag, Berlin, New York, 1985. doi: 10.1007/978-3-662-09922-3.  Google Scholar

[29]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[30]

V. H. Nguyen, The sharp Poincaré–Sobolev type inequalities in the hyperbolic spaces $\mathbb H^n$, J. Math. Anal. Appl., 462 (2018), 1570-1584.  doi: 10.1016/j.jmaa.2018.02.054.  Google Scholar

[31]

V. H. Nguyen, Improved Moser-Trudinger type inequalities in the hyperbolic space $\mathbb H^n$, Nonlinear Anal., 168 (2018), 67-80.  doi: 10.1016/j.na.2017.11.009.  Google Scholar

[32]

V. H. Nguyen, Improved Moser–Trudinger inequality of Tintarev type in dimension $n$ and the existence of its extremal functions, Ann. Global Anal. Geom., 54 (2018), 237-256.  doi: 10.1007/s10455-018-9599-z.  Google Scholar

[33]

V. H. Nguyen, Improved singular Moser-Trudinger inequalities and their extremal functions, Potential Anal., in press. Google Scholar

[34]

V. H. Nguyen, The sharp Hardy–Moser–Trudinger inequality in dimension $n$, preprint, arXiv: 1909.12587. Google Scholar

[35]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$] (Russian), Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.   Google Scholar

[36]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb R}^2$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[37]

A. Tertikas and C. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality, Ann. Mat. Pura Appl. (4), 186 (2007), 645-662. doi: 10.1007/s10231-006-0024-z.  Google Scholar

[38]

C. Tintarev, Trudinger–Moser inequality with remainder terms, J. Funct. Anal., 266 (2014), 55-66.  doi: 10.1016/j.jfa.2013.09.009.  Google Scholar

[39]

N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[40]

G. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Adv. Math., 230 (2012), 294-320.  doi: 10.1016/j.aim.2011.12.001.  Google Scholar

[41]

X. Wang, Improved Hardy-Adams inequality on hyperbolic space of dimension four, Nonlinear Anal., 182 (2019), 45-56.  doi: 10.1016/j.na.2018.12.007.  Google Scholar

[42]

X. Wang, Singular Hardy-Moser-Trudinger inequality and the existence of extremals on the unit disc, Commun. Pure Appl. Anal., 18 (2019), 2741-2757.  doi: 10.3934/cpaa.2019121.  Google Scholar

[43]

Y. Yang, A sharp form of the Moser–Trudinger inequality on a compact Riemannian surface, Trans. Amer. Math. Soc., 359 (2007), 5761-5776.  doi: 10.1090/S0002-9947-07-04272-9.  Google Scholar

[44]

Y. Yang and X. Zhu, An improved Hardy-Trudinger-Moser inequality, Ann. Global Anal. Geom., 49 (2016), 23-41.  doi: 10.1007/s10455-015-9478-9.  Google Scholar

[45]

Q. YangD. Su and Y. Kong, Sharp Moser-Trudinger inequalities on Riemannian manifolds with negative curvature, Ann. Mat. Pura Appl., 195 (2016), 459-471.  doi: 10.1007/s10231-015-0472-4.  Google Scholar

[46]

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

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