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Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents
Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
$ B $ |
$ \mathbb{R}^2 $ |
$ 0<t<2 $ |
$ u \in C_0^\infty({B}) $ |
$ \int_{{B}}|\nabla u|^2 dx- \int_{{B}}\frac{u^2}{(1-|x|^2)^2}dx\leq1, $ |
$ C_{0}>0 $ |
$ \int_{{B}}\frac{e^{4\pi(1-t/2)u^2}}{|x|^t} dx\leq C_{0}. $ |
$ t = 0 $ |
$ 0<t<2 $ |
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. |
[2] |
Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, Nolinear Differential Equations Application, 13 (2007), 585-603. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[9] |
M. Flucher,
Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 67 (1992), 471-497.
doi: 10.1007/BF02566514. |
[10] |
N. Lam,
Equivalence of sharp Trudinger-Moser-Adams inequalities, Commun. Pure Appl. Anal., 16 (2017), 973-997.
doi: 10.3934/cpaa.2017047. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
Y. Li,
Trudinger-Moser inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
G. Lu and M. Zhu, A sharp Trudinger-Moser type inequality involving $L ^n$ norm in the entire space $\mathbb{R}^n$. |
[31] |
G. Mancini and K. Sandeep,
Moser-Trudinger inequality on conformal discs, Commum. Contemp. Math., 12 (2010), 1055-1068.
doi: 10.1142/S0219199710004111. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[38] |
V. I. Yudovich,
Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.
|
[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. |
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. |
[2] |
Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, Nolinear Differential Equations Application, 13 (2007), 585-603. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[9] |
M. Flucher,
Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 67 (1992), 471-497.
doi: 10.1007/BF02566514. |
[10] |
N. Lam,
Equivalence of sharp Trudinger-Moser-Adams inequalities, Commun. Pure Appl. Anal., 16 (2017), 973-997.
doi: 10.3934/cpaa.2017047. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
Y. Li,
Trudinger-Moser inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
G. Lu and M. Zhu, A sharp Trudinger-Moser type inequality involving $L ^n$ norm in the entire space $\mathbb{R}^n$. |
[31] |
G. Mancini and K. Sandeep,
Moser-Trudinger inequality on conformal discs, Commum. Contemp. Math., 12 (2010), 1055-1068.
doi: 10.1142/S0219199710004111. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[38] |
V. I. Yudovich,
Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.
|
[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. |
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