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September  2019, 18(5): 2717-2733. doi: 10.3934/cpaa.2019121

Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc

Xumin Wang ,

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author

Received  October 2018 Revised  January 2019 Published  April 2019

Fund Project: The author was partly supported by grant from the NNSF of China (No.11371056).

We present the singular Hardy-Trudinger-Moser inequality and the existence of their extremal functions on the unit disc
$ B $
 in
$ \mathbb{R}^2 $
. As our first main result, we show that for any
$ 0<t<2 $
 and
$ u \in C_0^\infty({B}) $
 satisfying
$ \int_{{B}}|\nabla u|^2 dx- \int_{{B}}\frac{u^2}{(1-|x|^2)^2}dx\leq1, $
there exists a constant
$ C_{0}>0 $
 such that the following inequality holds
$ \int_{{B}}\frac{e^{4\pi(1-t/2)u^2}}{|x|^t} dx\leq C_{0}. $
Furthermore, by the method of blow-up analysis, we establish the existence of extremal functions in a suitable function space. Our results extend those in Wang and Ye [36] from the non-singular case
$ t = 0 $
 to the singular case for
$ 0<t<2 $
.
Keywords: Singular Hardy-Trudinger-Moser inequality, rearrangement, blow-up analysis.
Mathematics Subject Classification: Primary: 35J50; Secondary: 46E30, 46E35.
Citation: Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121
References:
[1]

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

[2]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, Nolinear Differential Equations Application, 13 (2007), 585-603. Google Scholar

[3]

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

[4]

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

[5]

L. Chen, J. Li, G. Lu and C. Zhang, Sharpened Adams inequality and ground state solutions to the bi-Laplacian equation in $R^4$, Adv. Nonlinear Stud., 18 (2018), 429-452.  doi: 10.1515/ans-2018-2020.  Google Scholar

[6]

M. Dong, N. Lam and G. Lu, Sharp weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg inequalities and their extremal functions, Nonlinear Anal., 173 (2018), 75-98.  doi: 10.1016/j.na.2018.03.006.  Google Scholar

[7]

M. Dong and G. Lu, Best constants and existence of maximizers for weighted Trudinger-Moser inequalities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 88, 26 pp. doi: 10.1007/s00526-016-1014-7.  Google Scholar

[8]

Y. Dong and Q. Yang, An interpolation of Hardy inequality and Moser-Trudinger inequality on Riemannian manifolds with negative curvature, Acta. Mathematica Sinica., English Series, 32 (2016), 856-866.  doi: 10.1007/s10114-016-5129-8.  Google Scholar

[9]

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

[10]

N. Lam, Equivalence of sharp Trudinger-Moser-Adams inequalities, Commun. Pure Appl. Anal., 16 (2017), 973-997.  doi: 10.3934/cpaa.2017047.  Google Scholar

[11]

N. Lam and G. Lu, Sharp constants and optimizers for a class of Caffarelli-Kohn-Nirenberg inequalities, Adv. Nonlinear Stud., 17 (2017), 457-480.  doi: 10.1515/ans-2017-0012.  Google Scholar

[12]

N. Lam and G. Lu, Sharp singular Trudinger-Moser-Adams type inequalities with exact growth, Geometric methods in PDE's, 43–80, Springer INdAM Ser., 13, Springer, Cham, 2015.  Google Scholar

[13]

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

[14]

N. Lam and G. Lu, Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math., 231 (2012), 3259-3287.  doi: 10.1016/j.aim.2012.09.004.  Google Scholar

[15]

J. Li, G. Lu and Q. Yang, Fourier analysis and optimal Hardy-Adams inequalities on hyperbolic spaces of any even dimension, Adv. Math., 33 (2018), 350-385.  doi: 10.1016/j.aim.2018.05.035.  Google Scholar

[16]

J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations, (2018), 57-84.  doi: 10.1007/s00526-018-1352-8.  Google Scholar

[17]

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

[18]

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

[19]

Y. Li, Remarks on the extremal functions for the Moser-Trudinger inequality, Acta Math. Sin. (Engl. Ser.), 22 (2006), 545-550.  doi: 10.1007/s10114-005-0568-7.  Google Scholar

[20]

Y. Li and C. Ndiaye, Extremal functions for Moser-Trudinger type inequality on compact closed 4-manifolds, J. Geom. Anal., 17 (2007), 669-699.  doi: 10.1007/BF02937433.  Google Scholar

[21]

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

[22]

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

[23]

G. Lu and H. Tang, Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud., 13 (2013), 1035-1052.  doi: 10.1515/ans-2013-0415.  Google Scholar

[24]

G. Lu and H. Tang, Sharp Moser-Trudinger inequalities on hyperbolic spaces with exact growth condition, J. Geom. Anal., 26 (2016), 837-857.  doi: 10.1007/s12220-015-9573-y.  Google Scholar

[25]

G. Lu and Q. Yang, A sharp Trudinger-Moser inequality on any bounded and convex plannar domain, Calc. Var. Partial Differential Equations, 55 (2016).  doi: 10.1007/s00526-016-1077-5.  Google Scholar

[26]

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

[27]

G. Lu and Q. Yang, Paneitz operators on hyperbolic spaces and higher order Hardy-Sobolev-Maz'ya inequalities on half spaces, Amer. J. Math., to appear. Google Scholar

[28]

G. Lu and Y. Yang, Adams' inequalities for bi-Laplacian and extremal functions in dimension four, Adv. Math., 220 (2009), 1135-1170.  doi: 10.1016/j.aim.2008.10.011.  Google Scholar

[29]

G. Lu and Y. Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979.  doi: 10.3934/dcds.2009.25.963.  Google Scholar

[30]

G. Lu and M. Zhu, A sharp Trudinger-Moser type inequality involving $L ^n$ norm in the entire space $\mathbb{R}^n$. Google Scholar

[31]

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

[32]

G. Mancini, K. Sandeep and K. Tintarev, Trudinger-Moser inequality in the hyperbolic spaces $\mathbb{H}^N$, Adv. Nonlinear Anal., 2 (2013), 309-324.  doi: 10.1515/anona-2013-0001.  Google Scholar

[33]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[34]

S. I. Pohozaev, The Sobolev embedding in the case pl = n, Proceeding of the Technical Scientific Conference on Advances of Scientific Research, 1964–1965. Mathematics Section, Moskov. Energet. Inst., (1965), 158–170. Google Scholar

[35]

N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[36]

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

[37]

Q. Yang, D. Su and Y. Kong, Sharp Moser-Trudinger inequalities on Riemannian manifolds with negative curvature, Annali di Matematica Pura ed Applicata, 195 (2016), 459-471.  doi: 10.1007/s10231-015-0472-4.  Google Scholar

[38]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.   Google Scholar

[39]

C. Zhang and L. Chen, Concentration-compactness principle of singular Trudinger-Moser inequalities in $R^n$ and n-Laplace equations, Adv. Nonlinear Stud., 18 (2018), 567-585.  doi: 10.1515/ans-2017-6041.  Google Scholar

show all references

References:
[1]

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

[2]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, Nolinear Differential Equations Application, 13 (2007), 585-603. Google Scholar

[3]

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

[4]

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

[5]

L. Chen, J. Li, G. Lu and C. Zhang, Sharpened Adams inequality and ground state solutions to the bi-Laplacian equation in $R^4$, Adv. Nonlinear Stud., 18 (2018), 429-452.  doi: 10.1515/ans-2018-2020.  Google Scholar

[6]

M. Dong, N. Lam and G. Lu, Sharp weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg inequalities and their extremal functions, Nonlinear Anal., 173 (2018), 75-98.  doi: 10.1016/j.na.2018.03.006.  Google Scholar

[7]

M. Dong and G. Lu, Best constants and existence of maximizers for weighted Trudinger-Moser inequalities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 88, 26 pp. doi: 10.1007/s00526-016-1014-7.  Google Scholar

[8]

Y. Dong and Q. Yang, An interpolation of Hardy inequality and Moser-Trudinger inequality on Riemannian manifolds with negative curvature, Acta. Mathematica Sinica., English Series, 32 (2016), 856-866.  doi: 10.1007/s10114-016-5129-8.  Google Scholar

[9]

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

[10]

N. Lam, Equivalence of sharp Trudinger-Moser-Adams inequalities, Commun. Pure Appl. Anal., 16 (2017), 973-997.  doi: 10.3934/cpaa.2017047.  Google Scholar

[11]

N. Lam and G. Lu, Sharp constants and optimizers for a class of Caffarelli-Kohn-Nirenberg inequalities, Adv. Nonlinear Stud., 17 (2017), 457-480.  doi: 10.1515/ans-2017-0012.  Google Scholar

[12]

N. Lam and G. Lu, Sharp singular Trudinger-Moser-Adams type inequalities with exact growth, Geometric methods in PDE's, 43–80, Springer INdAM Ser., 13, Springer, Cham, 2015.  Google Scholar

[13]

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

[14]

N. Lam and G. Lu, Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math., 231 (2012), 3259-3287.  doi: 10.1016/j.aim.2012.09.004.  Google Scholar

[15]

J. Li, G. Lu and Q. Yang, Fourier analysis and optimal Hardy-Adams inequalities on hyperbolic spaces of any even dimension, Adv. Math., 33 (2018), 350-385.  doi: 10.1016/j.aim.2018.05.035.  Google Scholar

[16]

J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations, (2018), 57-84.  doi: 10.1007/s00526-018-1352-8.  Google Scholar

[17]

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

[18]

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

[19]

Y. Li, Remarks on the extremal functions for the Moser-Trudinger inequality, Acta Math. Sin. (Engl. Ser.), 22 (2006), 545-550.  doi: 10.1007/s10114-005-0568-7.  Google Scholar

[20]

Y. Li and C. Ndiaye, Extremal functions for Moser-Trudinger type inequality on compact closed 4-manifolds, J. Geom. Anal., 17 (2007), 669-699.  doi: 10.1007/BF02937433.  Google Scholar

[21]

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

[22]

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

[23]

G. Lu and H. Tang, Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud., 13 (2013), 1035-1052.  doi: 10.1515/ans-2013-0415.  Google Scholar

[24]

G. Lu and H. Tang, Sharp Moser-Trudinger inequalities on hyperbolic spaces with exact growth condition, J. Geom. Anal., 26 (2016), 837-857.  doi: 10.1007/s12220-015-9573-y.  Google Scholar

[25]

G. Lu and Q. Yang, A sharp Trudinger-Moser inequality on any bounded and convex plannar domain, Calc. Var. Partial Differential Equations, 55 (2016).  doi: 10.1007/s00526-016-1077-5.  Google Scholar

[26]

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

[27]

G. Lu and Q. Yang, Paneitz operators on hyperbolic spaces and higher order Hardy-Sobolev-Maz'ya inequalities on half spaces, Amer. J. Math., to appear. Google Scholar

[28]

G. Lu and Y. Yang, Adams' inequalities for bi-Laplacian and extremal functions in dimension four, Adv. Math., 220 (2009), 1135-1170.  doi: 10.1016/j.aim.2008.10.011.  Google Scholar

[29]

G. Lu and Y. Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979.  doi: 10.3934/dcds.2009.25.963.  Google Scholar

[30]

G. Lu and M. Zhu, A sharp Trudinger-Moser type inequality involving $L ^n$ norm in the entire space $\mathbb{R}^n$. Google Scholar

[31]

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

[32]

G. Mancini, K. Sandeep and K. Tintarev, Trudinger-Moser inequality in the hyperbolic spaces $\mathbb{H}^N$, Adv. Nonlinear Anal., 2 (2013), 309-324.  doi: 10.1515/anona-2013-0001.  Google Scholar

[33]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[34]

S. I. Pohozaev, The Sobolev embedding in the case pl = n, Proceeding of the Technical Scientific Conference on Advances of Scientific Research, 1964–1965. Mathematics Section, Moskov. Energet. Inst., (1965), 158–170. Google Scholar

[35]

N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[36]

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

[37]

Q. Yang, D. Su and Y. Kong, Sharp Moser-Trudinger inequalities on Riemannian manifolds with negative curvature, Annali di Matematica Pura ed Applicata, 195 (2016), 459-471.  doi: 10.1007/s10231-015-0472-4.  Google Scholar

[38]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.   Google Scholar

[39]

C. Zhang and L. Chen, Concentration-compactness principle of singular Trudinger-Moser inequalities in $R^n$ and n-Laplace equations, Adv. Nonlinear Stud., 18 (2018), 567-585.  doi: 10.1515/ans-2017-6041.  Google Scholar

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