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

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