On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, p. 297-314
The full text of recent articles is available to journal subscribers only. See the article on the journal's website
In this article, we establish the weighted Trudinger–Moser inequality of the scaling invariant form including its best constant and prove the existence of a maximizer for the associated variational problem. The non-singular case was treated by Adachi and Tanaka (1999) [1] and the existence of a maximizer is a new result even for the non-singular case. We also discuss the relation between the best constants of the weighted Trudinger–Moser inequality and the Caffarelli–Kohn–Nirenberg inequality in the asymptotic sense.
DOI : https://doi.org/10.1016/j.anihpc.2013.03.004
Classification:  46E35,  35J20
Keywords: Weighted Trudinger–Moser inequality, Existence of maximizer, Caffarelli–Kohn–Nirenberg inequality with asymptotics
@article{AIHPC_2014__31_2_297_0,
     author = {Ishiwata, Michinori and Nakamura, Makoto and Wadade, Hidemitsu},
     title = {On the sharp constant for the weighted Trudinger--Moser type inequality of the scaling invariant form},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {2},
     year = {2014},
     pages = {297-314},
     doi = {10.1016/j.anihpc.2013.03.004},
     zbl = {1311.46034},
     mrnumber = {3181671},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_2_297_0}
}
Ishiwata, Michinori; Nakamura, Makoto; Wadade, Hidemitsu. On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, pp. 297-314. doi : 10.1016/j.anihpc.2013.03.004. http://www.numdam.org/item/AIHPC_2014__31_2_297_0/

[1] S. Adachi, K. Tanaka, A scale-invariant form of Trudinger–Moser inequality and its best exponent, Proc. Amer. Math. Soc. 1102 (1999), 148-153 | MR 1747573 | Zbl 0951.35525

[2] F.J. Almgren, E.H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), 683-773 | MR 1002633 | Zbl 0688.46014

[3] C. Bennett, R. Sharpley, Interpolation of Operators, Academic, New York (1988) | MR 928802 | Zbl 0647.46057

[4] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259-275 | Numdam | MR 768824 | Zbl 0563.46024

[5] D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in 2 , Comm. Partial Differential Equations 17 (1992), 407-435 | MR 1163431 | Zbl 0763.35034

[6] L. Carleson, S.-Y.A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2) 110 (1986), 113-127 | MR 878016 | Zbl 0619.58013

[7] J.L. Chern, C.S. Lin, Minimizers of Caffarelli–Kohn–Nirenberg inequalities on domains with the singularity on the boundary, Arch. Ration. Mech. Anal. 197 (2010), 401-432 | MR 2660516 | Zbl 1197.35091

[8] M. Flucher, Extremal functions for the Trudinger–Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (1992), 471-479 | MR 1171306 | Zbl 0763.58008

[9] N. Ghoussoub, X. Kang, Hardy–Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 767-793 | MR 2097030 | Zbl 1232.35064

[10] N. Ghoussoub, F. Robert, Concentration estimates for Emden–Fowler equations with boundary singularities and critical growth, IMRP Int. Math. Res. Pap. 21867 (2006), 1-85 | MR 2210661 | Zbl 1154.35049

[11] N. Ghoussoub, F. Robert, The effect of curvature on the best constant in the Hardy–Sobolev inequalities, Geom. Funct. Anal. 16 (2006), 1201-1245 | MR 2276538 | Zbl 1232.35044

[12] C.H. Hsia, C.S. Lin, H. Wadade, Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal. 259 (2010), 1816-1849 | MR 2665412 | Zbl 1198.35098

[13] M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in N , Math. Ann. 351 (2011), 781-804 | MR 2854113 | Zbl 1241.58007

[14] H. Kozono, T. Sato, H. Wadade, Upper bound of the best constant of a Trudinger–Moser inequality and its application to a Gagliardo–Nirenberg inequality, Indiana Univ. Math. J. 55 (2006), 1951-1974 | MR 2284552 | Zbl 1126.46023

[15] Y. Li, B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in n , Indiana Univ. Math. J. 57 (2008), 451-480 | MR 2400264 | Zbl 1157.35032

[16] K.C. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc. 348 (1996), 2663-2671 | MR 1333394 | Zbl 0861.49001

[17] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970), 1077-1092 | MR 301504 | Zbl 0203.43701

[18] S. Nagayasu, H. Wadade, Characterization of the critical Sobolev space on the optimal singularity at the origin, J. Funct. Anal. 258 (2010), 3725-3757 | MR 2606870 | Zbl 1209.46017

[19] T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrodinger equation, Nonlinear Anal. 14 (1990), 765-769 | MR 1049119 | Zbl 0715.35073

[20] T. Ogawa, T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrodinger mixed problem, J. Math. Anal. Appl. 155 (1991), 531-540 | MR 1097298 | Zbl 0733.35095

[21] T. Ozawa, Characterization of Trudinger's inequality, J. Inequal. Appl. 1 (1997), 369-374 | MR 1732633 | Zbl 0921.46023

[22] T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal. 127 (1995), 259-269 | MR 1317718 | Zbl 0846.46025

[23] B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in 2 , J. Funct. Anal. 219 (2005), 340-367 | MR 2109256 | Zbl 1119.46033

[24] M. Struwe, Critical points of embeddings of H 0 1,n into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), 425-464 | Numdam | MR 970849 | Zbl 0664.35022

[25] N.S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483 | MR 216286 | Zbl 0163.36402

[26] J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202 | MR 768629 | Zbl 0561.35003