Abstract
In this paper, we will study the Trudinger-Moser inequalities with the monomial weight \(\left| x_{1}\right| ^{A_{1}}...\left| x_{N}\right| ^{A_{N}}\) in \( \mathbb {R} ^{N}\) with \(A_{1}\ge 0,..., A_{N}\ge 0\). Moreover, we investigate the Trudinger-Moser inequalities on both domains of finite and infinite volume. More importantly, we will exhibit the best constants for our results. In the particular case \(A_{1}=\cdots =\) \(A_{N}=0\), we recover many results about the Trudinger-Moser inequalities without weight established in the literature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adachi, S., Tanaka, K.: Trudinger type inequalities in \(\mathbb{R}^{N}\) and their best exponents. Proc. Am. Math. Soc. 128, 2051–2057 (1999)
Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. of Math. (2) 128(2), 385–398 (1988)
Adimurthi, Yang, Y.: An interpolation of Hardy inequality and Trundinger-Moser inequality in \({\mathbb{R}}^{N}\) and its applications. Int. Math. Res. Not. 13, 2394–2426 (2010)
Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften, vol. 348. Springer, Berlin (2013)
Brezis, H.: Is there failure of the inverse function theorem? Morse theory, minimax theory and theirapplications to nonlinear differential equations. In: Proceedings of the Workshop held at the Chinese Academy of Sciences, Beijing, 1999, 23–33, New Stud. Adv. Math., 1, Int. Press, Somerville, MA, (2003)
Brezis, H., Vázquez, J.L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 10, 443–469 (1997)
Cabré, X., Ros-Oton, X.: Regularity of stable solutions up to dimension 7 in domains of double revolution. Commun. Partial Differ. Equ. 38, 135–154 (2013)
Cabré, X., Ros-Oton, X.: Sobolev and isoperimetric inequalities with monomial weights. J. Differ. Equ. 255(11), 4312–4336 (2013)
do Ó, J.M.: \(N-\)Laplacian equations in \({\mathbb{R}}^{N}\) with critical growth. Abstr. Appl. Anal. 2(3–4), 301–315 (1997)
Ibrahim, S., Masmoudi, N., Nakanishi, K.: Trudinger-Moser inequality on the whole plane with the exact growth condition. J. Eur. Math. Soc. 17, 819–835 (2015)
Lam, N.: Equivalence of sharp Trudinger-Moser-Adams inequalities. Commun. Pure Appl. Anal. 16(3), 973–997 (2017)
Lam, N., Lu, G., Tang, H.: Sharp affine and improved Moser-Trudinger-Adams type inequalities on unbounded domains in the spirit of Lions. J. Geom. Anal. 27(1), 300–334 (2017)
Lam, N., Lu, G., Zhang, L.: Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities. Rev. Mat. Iberoam, (to appear). arXiv:1504.04858
Li, Y.X., Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in \( \mathbb{R} ^{n}\). Indiana Univ. Math. J. 57(1), 451–480 (2008)
Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)
Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoam. 1(2), 45–121 (1985)
Lu, G., Tang, H.: Sharp Moser-Trudinger inequalities on hyperbolic spaces with exact growth condition. J. Geom. Anal. 26(2), 837–857 (2016)
Masmoudi, N., Sani, F.: Trudinger-Moser inequalities with the exact growth condition in \(\mathbb{R}^{N}\). Commun. Partial Differ. Equ. 40(8), 1408–1440 (2015)
Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/71)
Nguyen, V.H.: Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half-spaces via mass transport and consequences. Proc. Lond. Math. Soc. (3) 111(1), 127–148 (2015)
Pohožaev, S.I.: On the eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\). (Russian). Dokl. Akad. Nauk SSSR 165, 36–39 (1965)
Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in \( \mathbb{R} ^{2}\). J. Funct. Anal. 219(2), 340–367 (2005)
Talenti, G.: A weighted version of a rearrangement inequality. Ann. Univ. Ferr. 43, 121–133 (1997)
Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)
Judovič, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. (Russian). Dokl. Akad. Nauk SSSR 138, 805–808 (1961)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of this work was partially supported by the PIMS-Math Distinguished Post-doctoral Fellowship from the Pacific Institute for the Mathematical Sciences.
Rights and permissions
About this article
Cite this article
Lam, N. Sharp Trudinger-Moser inequalities with monomial weights. Nonlinear Differ. Equ. Appl. 24, 39 (2017). https://doi.org/10.1007/s00030-017-0456-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-017-0456-8