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.
@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}, volume = {31}, year = {2014}, pages = {297-314}, doi = {10.1016/j.anihpc.2013.03.004}, mrnumber = {3181671}, zbl = {1311.46034}, language = {en}, url = {http://dml.mathdoc.fr/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, Tome 31 (2014) pp. 297-314. doi : 10.1016/j.anihpc.2013.03.004. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_2_297_0/
[1] 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] Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), 683-773 | MR 1002633 | Zbl 0688.46014
, ,[3] Interpolation of Operators, Academic, New York (1988) | MR 928802 | Zbl 0647.46057
, ,[4] First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259-275 | Numdam | MR 768824 | Zbl 0563.46024
, , ,[5] Nontrivial solution of semilinear elliptic equation with critical exponent in , Comm. Partial Differential Equations 17 (1992), 407-435 | MR 1163431 | Zbl 0763.35034
,[6] 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] 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] Extremal functions for the Trudinger–Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (1992), 471-479 | MR 1171306 | Zbl 0763.58008
,[9] 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] 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] 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] Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal. 259 (2010), 1816-1849 | MR 2665412 | Zbl 1198.35098
, , ,[13] Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in , Math. Ann. 351 (2011), 781-804 | MR 2854113 | Zbl 1241.58007
,[14] 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] A sharp Trudinger–Moser type inequality for unbounded domains in , Indiana Univ. Math. J. 57 (2008), 451-480 | MR 2400264 | Zbl 1157.35032
, ,[16] Extremal functions for Moser's inequality, Trans. Amer. Math. Soc. 348 (1996), 2663-2671 | MR 1333394 | Zbl 0861.49001
,[17] A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970), 1077-1092 | MR 301504 | Zbl 0203.43701
,[18] 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] 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] 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] Characterization of Trudinger's inequality, J. Inequal. Appl. 1 (1997), 369-374 | MR 1732633 | Zbl 0921.46023
,[22] On critical cases of Sobolev's inequalities, J. Funct. Anal. 127 (1995), 259-269 | MR 1317718 | Zbl 0846.46025
,[23] A sharp Trudinger–Moser type inequality for unbounded domains in , J. Funct. Anal. 219 (2005), 340-367 | MR 2109256 | Zbl 1119.46033
,[24] Critical points of embeddings of into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), 425-464 | Numdam | MR 970849 | Zbl 0664.35022
,[25] On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483 | MR 216286 | Zbl 0163.36402
,[26] A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202 | MR 768629 | Zbl 0561.35003
,