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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[33]

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

[34]

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

[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. 

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[33]

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

[34]

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

[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. 

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[1]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121

[2]

Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963

[3]

Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110

[4]

Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212

[5]

Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505

[6]

Changliang Zhou, Chunqin Zhou. On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 847-881. doi: 10.3934/dcds.2020064

[7]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011

[8]

Nguyen Lam. Equivalence of sharp Trudinger-Moser-Adams Inequalities. Communications on Pure and Applied Analysis, 2017, 16 (3) : 973-998. doi: 10.3934/cpaa.2017047

[9]

Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378

[10]

Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure and Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006

[11]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1747-1756. doi: 10.3934/dcdss.2020452

[12]

Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455

[13]

Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031

[14]

Yamin Wang. On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4257-4268. doi: 10.3934/cpaa.2020191

[15]

Mengjie Zhang. Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1721-1735. doi: 10.3934/cpaa.2021038

[16]

Sami Aouaoui, Rahma Jlel. Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 781-813. doi: 10.3934/dcds.2021137

[17]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143

[18]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951

[19]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[20]

Xuefeng Zhao, Yong Li. A Moser theorem for multiscale mappings. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022037

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (216)
  • HTML views (102)
  • Cited by (0)

Other articles
by authors

[Back to Top]