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May  2017, 16(3): 973-998. doi: 10.3934/cpaa.2017047

Equivalence of sharp Trudinger-Moser-Adams Inequalities

Department of Mathematics, University of British Columbia, The Pacific Institute for the Mathematical Sciences, Vancouver, BC V6T1Z4, Canada

Received  October 2016 Revised  January 2017 Published  February 2017

Fund Project: Research of this work was partially supported by the PIMS-Math Distinguished Post-doctoral Fellowship from the Pacific Institute for the Mathematical Sciences

Improved Trudinger-Moser-Adams type inequalities in the spirit of Lions were recently studied in [21]. The main purpose of this paper is to prove the equivalence of these versions of the Trudinger-Moser-Adams type inequalities and to set up the relations of these Trudinger-Moser-Adams best constants. Moreover, using these identities, we will investigate the existence and nonexistence of the optimizers for some Trudinger-Moser-Adams type ineq

Citation: Nguyen Lam. Equivalence of sharp Trudinger-Moser-Adams Inequalities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 973-998. doi: 10.3934/cpaa.2017047
References:
[1]

S. Adachi and K. Tanaka, Trudinger type inequalities in ℝN 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. of Math., 128 (1988), 385-398. doi: 10.2307/1971445. Google Scholar

[3]

Adimurthi and K. Sandeep, A singular Trudinger-Moser embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603. doi: 10.1007/s00030-006-4025-9. Google Scholar

[4]

Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger-Moser inequality in ℝN and its applications, Int. Math. Res. Not. IMRN, 13 (2010), 2394-2426. Google Scholar

[5]

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

[6]

D. CassaniF. Sani and C. Tarsi, Equivalent Moser type inequalities in R2 and the zero mass case, J. Funct. Anal., 267 (2014), 4236-4263. doi: 10.1016/j.jfa.2014.09.022. Google Scholar

[7]

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

[8]

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

[9]

J. M. do Ó, N-Laplacian equations in ℝN with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315. doi: 10.1155/S1085337597000419. Google Scholar

[10]

M. Dong, N. Lam and G. Lu, Singular Trudinger-Moser inequalities, Caffarelli-KohnNirenberg inequalities and their extremal functions, preprint.Google Scholar

[11]

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

[12]

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

[13]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454. doi: 10.1007/BF02565828. Google Scholar

[14]

L. Fontana and C. Morpurgo, Sharp Adams and Moser-Trudinger inequalities on ℝN and other spaces of infinite measure, preprint, arXiv: 1504.04678.Google Scholar

[15]

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

[16]

M. IshiwataM. Nakamura and H. Wadade, On the sharp constant for the weighted TrudingerMoser type inequality of the scaling invariant form, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 297-314. doi: 10.1016/j.anihpc.2013.03.004. Google Scholar

[17]

N. Lam, Maximizers for the singular Trudinger-Moser inequalities in the subcritical cases, Proc. Amer. Math. Soc. , to appear.Google Scholar

[18]

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

[19]

N. Lam and G. Lu, Sharp singular Adams inequalities in high order Sobolev spaces, Methods Appl. Anal., 19 (2012), 243-266. doi: 10.4310/MAA.2012.v19.n3.a2. Google Scholar

[20]

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

[21]

N. LamG. Lu and H. Tang, Sharp affine and improved Moser-Trudinger-Adams type inequalities on unbounded domains in the spirit of Lions, J. Geom. Anal., (2016). doi: 10.1007/s12220-016-9682-2. Google Scholar

[22]

N. Lam, G. Lu and L. Zhang, Equivalence of critical and subcritical sharp Trudinger-MoserAdams inequalities, Rev. Mat. Iberoam., to appear, arXiv: 1504.04858.Google Scholar

[23]

N. Lam, G. Lu and L. Zhang, Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities, preprint.Google Scholar

[24]

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

[25]

Y. Li and C. B. 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

[26]

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

[27]

K. C. 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

[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]

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]

S. I. Pohožaev, On the eigenfunctions of the equation Δu +λf(u) = 0, (Russian) Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. Google Scholar

[31]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in ℝ2, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013. Google Scholar

[32]

B. Ruf and F. Sani, Sharp Adams-type inequalities in ℝn, Trans. Amer. Math. Soc., 365 (2013), 645-670. doi: 10.1090/S0002-9947-2012-05561-9. Google Scholar

[33]

E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N. J. 1970 xiv+290 pp. Google Scholar

[34]

C. Tarsi, Adams' inequality and limiting Sobolev embeddings into Zygmund spaces, Potential Anal., 37 (2012), 353-385. doi: 10.1007/s11118-011-9259-4. Google Scholar

[35]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. Google Scholar

[36]

V. I. Judovič, 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 ℝN 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. of Math., 128 (1988), 385-398. doi: 10.2307/1971445. Google Scholar

[3]

Adimurthi and K. Sandeep, A singular Trudinger-Moser embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603. doi: 10.1007/s00030-006-4025-9. Google Scholar

[4]

Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger-Moser inequality in ℝN and its applications, Int. Math. Res. Not. IMRN, 13 (2010), 2394-2426. Google Scholar

[5]

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

[6]

D. CassaniF. Sani and C. Tarsi, Equivalent Moser type inequalities in R2 and the zero mass case, J. Funct. Anal., 267 (2014), 4236-4263. doi: 10.1016/j.jfa.2014.09.022. Google Scholar

[7]

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

[8]

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

[9]

J. M. do Ó, N-Laplacian equations in ℝN with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315. doi: 10.1155/S1085337597000419. Google Scholar

[10]

M. Dong, N. Lam and G. Lu, Singular Trudinger-Moser inequalities, Caffarelli-KohnNirenberg inequalities and their extremal functions, preprint.Google Scholar

[11]

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

[12]

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

[13]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454. doi: 10.1007/BF02565828. Google Scholar

[14]

L. Fontana and C. Morpurgo, Sharp Adams and Moser-Trudinger inequalities on ℝN and other spaces of infinite measure, preprint, arXiv: 1504.04678.Google Scholar

[15]

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

[16]

M. IshiwataM. Nakamura and H. Wadade, On the sharp constant for the weighted TrudingerMoser type inequality of the scaling invariant form, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 297-314. doi: 10.1016/j.anihpc.2013.03.004. Google Scholar

[17]

N. Lam, Maximizers for the singular Trudinger-Moser inequalities in the subcritical cases, Proc. Amer. Math. Soc. , to appear.Google Scholar

[18]

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

[19]

N. Lam and G. Lu, Sharp singular Adams inequalities in high order Sobolev spaces, Methods Appl. Anal., 19 (2012), 243-266. doi: 10.4310/MAA.2012.v19.n3.a2. Google Scholar

[20]

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

[21]

N. LamG. Lu and H. Tang, Sharp affine and improved Moser-Trudinger-Adams type inequalities on unbounded domains in the spirit of Lions, J. Geom. Anal., (2016). doi: 10.1007/s12220-016-9682-2. Google Scholar

[22]

N. Lam, G. Lu and L. Zhang, Equivalence of critical and subcritical sharp Trudinger-MoserAdams inequalities, Rev. Mat. Iberoam., to appear, arXiv: 1504.04858.Google Scholar

[23]

N. Lam, G. Lu and L. Zhang, Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities, preprint.Google Scholar

[24]

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

[25]

Y. Li and C. B. 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

[26]

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

[27]

K. C. 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

[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]

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]

S. I. Pohožaev, On the eigenfunctions of the equation Δu +λf(u) = 0, (Russian) Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. Google Scholar

[31]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in ℝ2, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013. Google Scholar

[32]

B. Ruf and F. Sani, Sharp Adams-type inequalities in ℝn, Trans. Amer. Math. Soc., 365 (2013), 645-670. doi: 10.1090/S0002-9947-2012-05561-9. Google Scholar

[33]

E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N. J. 1970 xiv+290 pp. Google Scholar

[34]

C. Tarsi, Adams' inequality and limiting Sobolev embeddings into Zygmund spaces, Potential Anal., 37 (2012), 353-385. doi: 10.1007/s11118-011-9259-4. Google Scholar

[35]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. Google Scholar

[36]

V. I. Judovič, 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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