Abstract
In this paper, we study on \(\mathbb {R}^{2}\) some new types of the sharp subcritical and critical Trudinger-Moser inequality that have close connections to the study of the optimizers for the classical Trudinger-Moser inequalities. For instance, one of our results can be read as follows: Let 0 ≤ β < 2, p ≥ 0, α ≥ 0. Then
if and only if α < 4π or α = 4π, p ≥ 2. The attainability and inattainability of these sharp inequalties will be also investigated using a new approach, namely the relations between the supremums of the sharp subcritical and critical ones. This new method will enable us to compute explicitly the supremums of the subcritical Trudinger-Moser inequalities in some special cases. Also, a version of Concentration-compactness principle in the spirit of Lions ( Lions, I. Rev. Mat. Iberoam. 1(1) 145–01 1985) will also be studied.
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Adachi, S., Tanaka, K.: Trudinger type inequalities in \(\mathbb {R}^{N}\) and their best exponents. Proc. Amer. Math. Soc. 128, 2051–2057 (1999)
Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. (2) 128(2), 385–398 (1988)
Adimurthi; Druet, O.: Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality. Comm. Partial Differ. Equat. 29(1-2), 295–322 (2004)
Adimurthi; Yang, Y.: An interpolation of Hardy inequality and Trudinger-Moser inequality in \(\mathbb {R}^{N}\) and its applications. Int. Math. Res. Not IMRN 13, 2394–2426 (2010)
Beckner, W.: Estimates on Moser embedding. Potential Anal. 20(4), 34–359 (2004)
Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82(4), 313–345 (1983)
Cao, D.: Nontrivial solution of semilinear elliptic equation with critical exponent in \(\mathbb {R}^{2}\). Comm. Partial Differ. Equat. 17(3-4), 407–435 (1992)
Carleson, L., Chang, S.Y.A.: On the existence of an extremal function for an inequality of. J. Moser. Bull. Sci. Math. (2) 110(2), 113–127 (1986)
Cassani, D., Sani, F., Tarsi, C.: Equivalent Moser type inequalities in \( \mathbb {R}^{2}\) and the zero mass case. J. Funct. Anal. 267(11), 4236–4263 (2014)
Černý, R., Cianchi, A., Hencl, S.: Concentration-compactness principles for Moser-Trudinger inequalities: new results and proofs. Ann. Mat. Pura Appl. (4) 192 (2), 225–243 (2013)
Cohn, W.S., Lu, G.: Best constants for Moser-Trudinger inequalities on the Heisenberg group. Indiana Univ. Math. J. 50(4), 1567–1591 (2001)
Cohn, W.S., Lu, G.: Sharp constants for Moser-Trudinger inequalities on spheres in complex space \(\mathbb {C}^{n}\). Comm. Pure Appl. Math. 57(11), 1458–1493 (2004)
Dong, M., Lu, G.: Best constants and existence of maximizers for weighted Moser-Trudinger inequalities. arXiv:1504.04847
Druet, O.: Multibumps analysis in dimension 2: quantification of blow-up levels. Duke Math. J. 132(2), 217–269 (2006)
De Figueiredo, D.G., Do Ó, J.M., Ruf, B.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Comm. Pure Appl. Math. 55(2), 135–152 (2002)
Flucher, M.: Extremal functions for the Trudinger-Moser inequality in 2 dimensions. Comment. Math. Helv. 67(3), 471–497 (1992)
Ibrahim, S., Masmoudi, N., Nakanishi, K.: Trudinger-moser inequality on the whole plane with the exact growth condition. J. Eur. Math. Soc. (JEMS) 17(4), 819–835 (2015)
Ishiwata, M.: nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in \(\mathbb {R}^{N}\). Math Existence Ann. 351(4), 781–804 (2011)
Ishiwata, M., Nakamura, M., Wadade, H.: On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form. Ann. I. H. Poincaré-AN 31, 297–314 (2014)
Lam, N., Lu, G.: Sharp Adams type inequalities in Sobolev spaces \(W^{m,\frac {n}{m}}\left (\mathbb {R}^{n}\right )\) for arbitrary integer m. J. Differ. Equat. 253, 1143–1171 (2012)
Lam, N., Lu, G.: Sharp singular Adams inequalities in high order Sobolev spaces. Methods Appl. Anal. 19(3), 243–266 (2012)
Lam, N., Lu, G.: Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications. Adv. Math. 231(6), 3259–3287 (2012)
Lam, N., Lu, G.: A new approach to sharp Trudinger-Moser and Adams type inequalities: a rearrangement-free argument. J. Differ. Equat. 255(3), 298–325 (2013)
Lam, N., Lu, G., Zhang, L.: Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities. arXiv:1504.04858
Lam, N., Lu, G., Zhang, L.: Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities. Preprint 2015
Li, Y.X.: Moser-Trudinger inequality on compact Riemannian manifolds of dimension two. J. Partial Differ. Equat. 14(2), 163–192 (2001)
Li, Y.X.: Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds. Sci. China Ser. A 48(5), 618–648 (2005)
Li, Y.X., Ruf, B.: Sharp Moser-Trudinger type inequality for unbounded domains in \(\mathbb {R}^{N}\). Indiana Univ. Math. J. 57(1), 451–480 (2008)
Lieb, E.H., Loss, M.: Analysis, 2nd edn., p xxii+346. Graduate Studies in Mathematics. 14. American Mathematical Society, Providence, RI (2001)
Lin, K.: Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc. 7, 2663–2671 (1996)
Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1(1), 145–01 (1985)
Lu, G., Tang, H.: Sharp Moser-Trudinger inequalities on hyperbolic spaces with the exact growth condition. J. Geom. Anal. 26(2), 837-857 (2016)
Lu, G., Yang, Y.: Adams’ inequalities for bi-Laplacian and extremal functions in dimension four. Adv. Math. 220, 1135–1170 (2009)
Masmoudi, N., Sani, F.: Trudinger-moser inequalities with the exact growth condition in \(\mathbb {R}^{N}\) and applications. Comm. Partial Differ. Equat. 40(8), 1408–1440 (2015)
Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20(1970/71), 1077–1092
Do Ó, J.M.: N−Laplacian equations in \(\mathbb {R}^{N}\) with critical growth. Abstr. Appl. Anal. 2(3-4), 301–315 (1997)
Pohožaev, S.I.: On the eigenfunctions of the equation Δu + 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)
Ruf, B., Sani, F.: Sharp Adams-type inequalities in \(\mathbb {R}^{N}\). Trans. Amer. Math. Soc. 365(2), 645–670 (2013)
Tarsi, C.: Adams’ inequality and limiting sobolev embeddings into zygmund spaces. Potential Anal. 37(4), 353–385 (2012)
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)
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Research of this work was partly supported by an AMS-Simons Travel Grant.
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Lam, N. Sharp Subcritical and Critical Trudinger-Moser Inequalities on \(\mathbb {R}^{2}\) and their Extremal Functions. Potential Anal 46, 75–103 (2017). https://doi.org/10.1007/s11118-016-9572-z
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DOI: https://doi.org/10.1007/s11118-016-9572-z
Keywords
- Trudinger-Moser inequality
- Unbounded domains
- Critical growth
- Extremal function
- Sharp constants
- Concentration compactness.